Statistics of spike trains in conductance-based neural networks: Rigorous results

The Journal of Mathematical Neuroscience20111:8

DOI: 10.1186/2190-8567-1-8

Received: 26 January 2011

Accepted: 25 August 2011

Published: 25 August 2011

Abstract

We consider a conductance-based neural network inspired by the generalized Integrate and Fire model introduced by Rudolph and Destexhe in 1996. We show the existence and uniqueness of a unique Gibbs distribution characterizing spike train statistics. The corresponding Gibbs potential is explicitly computed. These results hold in the presence of a time-dependent stimulus and apply therefore to non-stationary dynamics.

1 Introduction

Neural networks have an overwhelming complexity. While an isolated neuron can exhibit a wide variety of responses to stimuli [1], from regular spiking to chaos [2, 3], neurons coupled in a network via synapses (electrical or chemical) may show an even wider variety of collective dynamics [4] resulting from the conjunction of non-linear effects, time propagation delays, synaptic noise, synaptic plasticity, and external stimuli [5]. Focusing on the action potentials, this complexity is manifested by drastic changes in the spikes activity, for instance when switching from spontaneous to evoked activity (see for example A. Riehle’s team experiments on the monkey motor cortex [69]). However, beyond this, complexity may exist some hidden laws ruling an (hypothetical) “neural code” [10].

One way of unraveling these hidden laws is to seek some regularities or reproducibility in the statistics of spikes. While early investigations on spiking activities were focusing on firing rates where neurons are considered as independent sources, researchers concentrated more recently on collective statistical indicators such as pairwise correlations. Thorough experiments in the retina [11, 12] as well as in the parietal cat cortex [13] suggested that such correlations are crucial for understanding spiking activity. Those conclusions where obtained using the maximal entropy principle[14]. Assume that the average value of observables quantities (e.g., firing rate or spike correlations) has been measured. Those average values constitute constraints for the statistical model. In the maximal entropy principle, assuming stationarity, one looks for the probability distribution which maximizes the statistical entropy given those constraints. This leads to a (time-translation invariant) Gibbs distribution. In particular, fixing firing rates and the probability of pairwise coincidences of spikes lead to a Gibbs distribution having the same form as the Ising model. This idea has been introduced by Schneidman et al. in [11] for the analysis of retina spike trains. They reproduce accurately the probability of spatial spiking pattern. Since then, their approach has known a great success (see, e.g., [1517]), although some authors raised solid objections on this model [12, 1820] while several papers have pointed out the importance of temporal patterns of activity at the network level [2123]. As a consequence, a few authors [13, 24, 25] have attempted to define time-dependent models of Gibbs distributions where constraints include time-dependent correlations between pairs, triplets, and so on [26]. As a matter of fact, the analysis of the data of [11] with such models describes more accurately the statistics of spatio-temporal spike patterns [27].

Taking into account all constraints inherent to experiments, it seems extremely difficult to find an optimal model describing spike trains statistics. It is in fact likely that there is not one model, but many, depending on the experiment, the stimulus, the investigated part of the nervous system and so on. Additionally, the assumptions made in the works quoted above are difficult to control. Especially, the maximal entropy principle assumes a stationary dynamics while many experiments consider a time-dependent stimulus generating a time-dependent response where the stationary approximation may not be valid. At this stage, having an example where one knows the explicit form of the spike trains, probability distribution would be helpful to control those assumptions and to define related experiments.

This can be done considering neural network models. Although, to be tractable, such models may be quite away from biological plausibility, they can give hints on which statistics can be expected in real neural networks. But, even in the simplest examples, characterizing spike statistics arising from the conjunction of non-linear effects, time propagation delays, synaptic noise, synaptic plasticity, and external stimuli is far from being trivial on mathematical grounds.

In [28], we have nevertheless proposed an exact and explicit result for the characterization of spike trains statistics in a discrete-time version of Leaky Integrate-and-Fire neural network. The results were quite surprising. It has been shown that whatever the parameters value (in particular synaptic weights), spike trains are distributed according to a Gibbs distribution whose potential can be explicitly computed. The first surprise lies in the fact that this potential has infinite range, namely spike statistics has an infinite memory. This is because the membrane potential evolution integrates its past values and the past influence of the network via the leak term. Although leaky integrate and fire models have a reset mechanism that erases the memory of the neuron whenever it spikes, it is not possible to upper bound the next time of firing. As a consequence, statistics is non-Markovian (for recent examples of non-Markovian behavior in neural models see also [29]). The infinite range of the potential corresponds, in the maximal entropy principle interpretation, to having infinitely many constraints.

Nevertheless, the leak term influence decays exponentially fast with time (this property guarantees the existence and uniqueness of a Gibbs distribution). As a consequence, one can approximate the exact Gibbs distribution by the invariant probability of a Markov chain, with a memory depth proportional to the log of the (discrete time) leak term. In this way, the truncated potential corresponds to a finite number of constraints in the maximal entropy principle interpretation. However, the second surprise is that this approximated potential is nevertheless far from the Ising model or any of the models discussed above, which appear as quite bad approximations. In particular, there is a need to consider n-uplets of spikes with time delays. This mere fact asks hard problems about evidencing such type of potentials in experiments. Especially, new type of algorithms for spike trains analysis has to be developed [30].

The model considered in [28] is rather academic: time evolution is discrete, synaptic interactions are instantaneous, dynamics is stationary (the stimulus is time-constant) and, as in a leaky integrate and fire model, conductances are constant. It is therefore necessary to investigate whether our conclusions remain for more realistic neural networks models. In the present paper, we consider a conductance-based model introduced by Rudolph and Destexhe in [31] called “generalized Integrate and Fire” (gIF) model. This model allows one to consider realistic synaptic responses and conductances depending on spikes arising in the past of the network, leading to a rather complex dynamics which has been characterized in [32] in the deterministic case (no noise in the dynamics). Moreover, the biological plausibility of this model is well accepted [33, 34].

Here, we analyze spike statistics in the gIF model with noise and with a time-dependent stimulus. Moreover, the post-synaptic potential profiles are quite general and summarize all the examples that we know in the literature. Our main result is to prove the existence and uniqueness of a Gibbs measure characterizing spike trains statistics, for all parameters compatible with physical constraints (finite synaptic weights, bounded stimulus, and positive conductances). Here, as in [28], the corresponding Gibbs potential has infinite range corresponding to a non-Markovian dynamics, although Markovian approximations can be proposed in the gIF model too. The Gibbs potential depends on all parameters in the model (especially connectivity and stimulus) and has a form quite more complex than Ising-like models. As a by-product of the proof of our main result, additional interesting notions and results are produced such as continuity, with respect to a raster, or exponential decay of memory thanks to the shape of synaptic responses.

The paper is organized as follows. In Section 2, we briefly introduce integrate and fire models and propose two important extensions of the classical models: the spike has a duration and the membrane potential is reset to a non-constant value. These extensions, which are necessary for the validity of our mathematical results, render nevertheless the model more biologically plausible (see Section 9). One of the keys of the present work is to consider spike trains (raster plots) as infinite sequences. Since in gIF models, conductances are updated upon the occurrence of spikes, one has to consider two types of variables with distinct type of dynamics. On the one hand, the membrane potential, which is the physical variable associated with neurons dynamics, evolves continuously. On the other hand, spikes are discrete events. Conductances are updated according to these discrete-time events. The formalism introduced in Sections 2 and 3 allows us to handle properly this mixed dynamics. As a consequence, these sections define gIF model with more mathematical structure than the original paper [31] and mostly contain original results. Moreover, we add to the model several original features such as the consideration of a general form of synaptic profile with exponential decay or the introduction of noise. Section 4 proposes a preliminary analysis of gIF model dynamics. In Sections 5 and 6, we provide several useful mathematical propositions as a necessary step toward the analysis of spike statistics, developed in Section 7, where we prove the main result of the paper: existence and uniqueness of a Gibbs distribution describing spike statistics. Sections 8 and 9 are devoted to a discussion on practical consequences of our results for neuroscience.

2 Integrate and fire model

We consider the evolution of a set of N neurons. Here, neurons are considered as “points” instead of spatially extended and structured objects. As a consequence, we define, for each neuron k { 1 , , N } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq1_HTML.gif, a variable V k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq2_HTML.gif called the “membrane potential of neuron k at time t” without specification of which part of a real neuron (axon, soma, dendritic spine, …) it corresponds to. Denote V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq3_HTML.gif the vector ( V k ( t ) ) k = 1 N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq4_HTML.gif.

We focus here on “integrate and fire models”, where dynamics always consists of two regimes.

2.1 The “integrate regime”

Fix a real number θ called the “firing threshold of the neuron”.1 Below the threshold, V k < θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq5_HTML.gif, neuron k’s dynamics is driven by an equation of the form:
C k d V k d t + g k V k = i k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ1_HTML.gif
(1)

where C k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq6_HTML.gif is the membrane capacity of neuron k. In its most general form, the neuron k’s membrane conductance g k > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq7_HTML.gif depends on V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif plus additional variables such as the probability of having ionic channels open (see, e.g., Hodgkin-Huxley equations [35]) as well as on time t. The explicit form of g k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq9_HTML.gif in the present model is developed in Section 3.4. The current i k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq10_HTML.gif typically depends on time t and on the past activity of the network. It also contains a stochastic component modeling noise in the system (e.g., synaptic transmission, see Section 3.5).

2.2 LIF model

A classical example of integrate and fire model is the Leaky Integrate and Fire’s (LIF) introduced in [36] where Equation (1) reads:
d V k d t = V k τ L + i k ( t ) C k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ2_HTML.gif
(2)

where g k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq9_HTML.gif is a constant and τ L = C k g k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq11_HTML.gif is the characteristic time for membrane potential decay when no current is present (“leak term”).

2.3 Spikes

The dynamical evolution (1) may eventually lead V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif to exceed θ. If, at some time t, V k ( t ) = θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq12_HTML.gif, then neuron k emits a spike or “fires”. In our model, like in biophysics, a spike has a finite duration δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq13_HTML.gif; this is a generalization of the classical formulation of integrate and fire models where the spike is considered instantaneous. On biophysical grounds, δ is of order of a millisecond. Changing the time units, we may set δ = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq14_HTML.gif without loss of generality. Additionally, neurons have a refractory period τ refr > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq15_HTML.gif where they are not able to emit a new spike although their membrane potential can fluctuate below the threshold (see Figure 1). Hence, spikes emitted by a given neuron are separated by a minimal time scale
τ sep = δ + τ refr . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ3_HTML.gif
(3)
http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Fig1_HTML.jpg
Fig. 1

Time course of the membrane potential in our model. The blue dashed curve illustrates the shape of a real spike, but what we model is the red curve.

2.4 Raster plots

In experiments, spiking neuron activity is represented by “raster plots”, namely a graph with time in abscissa, and a neuron labeling in ordinate such that a vertical bar is drawn each “time” a neuron emits a spike. Since spikes have a finite duration δ such a representation limits the time resolution: events with a time scale smaller than δ are not distinguished. As a consequence, if neuron 1 fires at time t 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq16_HTML.gif and neuron 2 at time t 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq17_HTML.gif with | t 2 t 1 | < δ = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq18_HTML.gif the two spikes appear to be simultaneous on the raster. Thus, the raster representation introduces a time quantization and has a tendency to enhance synchronization. In gIF models, conductances are updated upon the occurrence of spikes (see Section 3.2) which are considered as such discrete events. This could correspond to the following “experiment”. Assume that we measure the spikes emitted by a set of in vitro neurons and that we use this information to update the conductances of a model, in order to see how this model “matches” the real neurons (see [34] for a nice investigation in this spirit). Then, we would have to take into account that the information provided by the experimental raster plot is discrete, even if the membrane potential evolves continuously. The consequences of this time discretization as well as the limit δ 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq19_HTML.gif are developed in the discussion section.

As a consequence, one has to consider two types of variables with distinct type of dynamics. On the one hand, the membrane potential, which is the physical variable associated with neuron dynamics, evolves with a continuous time. On the other hand, spikes, which are the quantities of interest in the present paper, are discrete events. To properly define this mixed dynamics and study its properties, we have to model spikes times and raster plots.

2.5 Spike times

If, at time t, V k ( t ) = θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq20_HTML.gif, a spike is registered at the integer time immediately after t, called the spike time. Choosing integers for the spike time occurrence is a direct consequence of setting δ = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq14_HTML.gif. Thus, to each neuron k and integer n, we associate a “spiking state” defined by:
ω k ( n ) = { 1 if t ] n 1 , n ] s u c h t h a t V k ( t ) = θ ; 0 otherwise. http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equa_HTML.gif

For convenience and in order to simplify the notations in the mathematical developments, we call [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq21_HTML.gif the largest integer which is ≤t (thus [ 1.2 ] = 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq22_HTML.gif and [ 1.2 ] = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq23_HTML.gif). Thus, the integer immediately after t is [ t + 1 ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq24_HTML.gif and we have therefore that ω k ( [ t + 1 ] ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq25_HTML.gif whenever V k ( t ) = θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq12_HTML.gif. Although, characteristic events in a raster plot are spikes (neuron fires), it is useful in subsequent developments to consider also the case when neuron is not firing ( ω k ( n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq26_HTML.gif).

2.6 Reset

In the classical formulation of integrate and fire models, the spike occurs simultaneously with a reset of the membrane potential to some constant value V reset http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq27_HTML.gif, called the “reset potential”. Instantaneous reset is a source of pathologies as discussed in [32, 37] and in the discussion section. Here, we consider that reset occurs after the time delay τ sep 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq28_HTML.gif including spike duration and refractory period. We set:
V k ( t ) = θ V k ( [ t + τ sep ] ) = V reset . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ4_HTML.gif
(4)

The reason why the reset time is the integer number [ t + τ sep ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq29_HTML.gif instead of the real t + τ sep http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq30_HTML.gif is that it eases the notations and proofs. Since the reset value is random (see below and Figure 1), this assumption has no impact on the dynamics.

Indeed, in our model, the reset value V reset http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq27_HTML.gif is not a constant. This is a Gaussian random variable with mean zero (we set the rest potential to zero without loss of generality) and variance σ R 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq31_HTML.gif. In this way, we model the spike duration and refractory period, as well as the random oscillations of the membrane potential during the refractory period. As a consequence, the value of V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif when the neuron can fire again is not a constant, as it is in classical IF models. A related reference (spiking neurons with partial reset) is [38]. The assumption that σ R 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq31_HTML.gif is necessary for our mathematical developments (see the bounds (37)). We assume σ R 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq32_HTML.gif to be small to avoid trivial and unrealistic situations where V reset θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq33_HTML.gif with a large probability leading the neuron to fire all the time. Note, however, that this is not a required assumption to establish our mathematical results. We also assume that, in successive resets, the random variables V reset http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq27_HTML.gif are independent.

2.7 The shape of membrane potential during the spike

On biophysical grounds, the time course of the membrane potential during the spike includes a depolarization and re-polarization phase due to the non-linear effects of gated ionic channels on the conductance. This leads to introduce, in modeling, additional variables such as activation/inactivation probabilities as in the Hodgkin-Huxley model [35] or adaptation current as, e.g., in FitzHugh-Nagumo model [3942] (see the discussion section for extensions of our results to those models). Here, since we are considering only one variable for the neuron state, the membrane potential, we need to define the spike profile, i.e., the course of V k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq2_HTML.gif during the time interval ( t , [ t + τ sep ] ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq34_HTML.gif. It turns out that the precise shape of this profile plays no role in the developments proposed here, where we concentrate on spike statistics. Indeed, a spike is registered whenever V k ( t ) = θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq20_HTML.gif, and this does not depend on the spike shape. What we need is therefore to define the membrane potential evolution before the spike, given by (1), and after the spike, given by (4) (see Figure 1).

2.8 Mathematical representation of raster plots

The “spiking pattern” of the neural network at integer time n is the vector ω ( n ) = ( ω k ( n ) ) k = 1 N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq35_HTML.gif. For m < n http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq36_HTML.gif, we note ω m n = { ω ( m ) ω ( m + 1 ) ω ( n ) } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq37_HTML.gif the ordered sequence of spiking patterns between m and n. Such sequences are called spike blocks. Additionally, we note that ω m n 1 1 ω n 1 n = ω m n http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq38_HTML.gif, the concatenation of the blocks ω m n 1 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq39_HTML.gif and ω n 1 n http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq40_HTML.gif.

Call A = { 0 , 1 } N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq41_HTML.gif the set of spiking patterns (alphabet). An element of X = def A Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq42_HTML.gif, i.e., a bi-infinite ordered sequence ω = { ω ( n ) } n = + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq43_HTML.gif of spiking patterns, is called a “raster plot”. It tells us which neurons are firing at each time n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif. In experiments, raster plots are obviously finite sequences of spiking pattern but the extension to Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq45_HTML.gif, especially the possibility of considering an arbitrary distant past (negative times), is a key of the present work. In particular, the notation ω n http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq46_HTML.gif refers to spikes occurring from −∞ to n.

To each raster ω X http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq47_HTML.gif and each neuron index j = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq48_HTML.gif, we associate an ordered (generically infinite) list of “spike times” { t j ( r ) ( ω ) } r = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq49_HTML.gif (integer numbers) such that t j ( r ) ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq50_HTML.gif is the r-th time of firing of neuron j in the raster ω. In other words, we have ω j ( n ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq51_HTML.gif if and only if n = t j ( r ) ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq52_HTML.gif for some r = 1 , , + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq53_HTML.gif. We use here the following convention. The index k is used for a post-synaptic neuron while the index j refers to pre-synaptic neurons. Spiking events are used to update the conductance of neuron k according to spikes emitted by pre-synaptic neurons. That is why we label the spike times with an index j.

We introduce here two specific rasters which are of use in the paper. We note Ω0 the raster such that ω k ( n ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq26_HTML.gif, k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq54_HTML.gif, n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq55_HTML.gif (no neuron ever fires) and Ω1 the raster ω k ( n ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq56_HTML.gif, i = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq57_HTML.gif, n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq55_HTML.gif (each neuron is firing at every integer time).

Finally, we use the following notation borrowed from [43]. We note, for n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif m 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq58_HTML.gif, and r integer:
ω = m , n ω if ω ( r ) = ω ( r ) , r { n m , , n } . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ5_HTML.gif
(5)

For simplicity, we consider that τ ref http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq59_HTML.gif, the refractory period, is smaller than 1 so that a neuron can fire two consecutive time steps (i.e., one can have ω k ( n ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq56_HTML.gif and ω k ( n + 1 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq60_HTML.gif). This constraint is discussed in Section 9.2.

2.9 Representation of time-dependent functions

Throughout the paper, we use the following convention. For a real function of t and ω, we write f ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq61_HTML.gif for f ( t , ω [ t ] ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq62_HTML.gif to simplify notations. This notation takes into account the duality between variables such as membrane potential evolving with respect to a continuous time and raster plots labeled with discrete time. Thus, the function f ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq61_HTML.gif is a function of the continuous variable t and of the spike block ω [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq63_HTML.gif, where by definition [ t ] t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq64_HTML.gif, namely f ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq61_HTML.gif depends on the spike sequences occurring before t. This constraint is imposed by causality.

2.10 Last reset time

We define τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif as the last time before t where neuron k’s membrane potential has been reset, in the raster ω. This is −∞ if the membrane potential has never been reset. As a consequence of our choice (4) for the reset time τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif is an integer number fixed by t and the raster before t. The membrane potential value of neuron k at time t is controlled by the reset value V reset http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq27_HTML.gif at time τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif and by the further subthreshold evolution (1) from time τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif to time t.

3 Generalized integrate and fire models

In this paper, we concentrate on an extension of (2), called “generalized Integrate-and-Fire” (gIF), introduced in [31], closer to biology [33, 34], since it considers more seriously neurons interactions via synaptic responses.

3.1 Synaptic conductances

Depending on the neurotransmitter they use for synaptic transmission (AMPA, NMDA, GABA A, GABA B [44]), neurons can be excitatory (population E http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq66_HTML.gif) or inhibitory (population I http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq67_HTML.gif). This is modeled by introducing reversal potentials E + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq68_HTML.gif for excitatory (typically E + 0 mV http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq69_HTML.gif for AMPA and NMDA) and E http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq70_HTML.gif for inhibitory ( E 70 mV http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq71_HTML.gif for GABA A and E 95 mV http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq72_HTML.gif for GABA B). We focus here on one population of excitatory and one population of inhibitory neurons although extensions to several populations may be considered as well. Also, each neuron is submitted to a current I k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq73_HTML.gif. We assume that this current has some stochastic component that mimics synaptic noise (Section 3.5).

The variation in the membrane potential of neuron k at time t reads:
C k d V k d t = g L , k ( V k E L ) g k ( E ) ( t ) ( V k E + ) g k ( I ) ( t ) ( V k E ) + I k ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ6_HTML.gif
(6)
where g L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq74_HTML.gif is a leak conductance, E L < 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq75_HTML.gif is the leak reversal potential (about −65 mV), g k ( E ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq76_HTML.gif the conductance of the excitatory population and g k ( I ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq77_HTML.gif the conductance of inhibitory population. They are given by:
g k ( E ) ( t ) = j E g k j ( t ) ; g k ( I ) ( t ) = j I g k j ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ7_HTML.gif
(7)

where g k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq78_HTML.gif is the conductance of the synaptic contact j k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq79_HTML.gif.

We may rewrite Equation (6) in the form (1) setting
g k ( t ) = g L , k + g k ( E ) ( t ) + g k ( I ) ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equb_HTML.gif
and
i k ( t ) = g L , k E L + g k ( E ) ( t ) E + + g k ( I ) ( t ) E + I k ( t ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equc_HTML.gif

3.2 Conductance update upon a spike occurrence

The conductances g k j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq80_HTML.gif in (7) depend on time t but also on pre-synaptic spikes occurring before t. This is a general statement, which is modeled in gIF models as follows. Upon arrival of a spike in the pre-synaptic neuron j at time t j ( r ) ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq50_HTML.gif, the membrane conductance of the post-synaptic neuron k is modified as:
g k j ( t ) = g k j ( t j ( r ) ( ω ) ) + G k j α k j ( t t j ( r ) ( ω ) ) , t > t j ( r ) ( ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ8_HTML.gif
(8)

In this equation, the quantity G k j 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq81_HTML.gif characterizes the maximal amplitude of the conductance during a post-synaptic potential. We use the convention that G k j = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq82_HTML.gif if and only if there is no synapse between j and k. This allows us to encode the graph structure of the neural network in the matrix G with entries G k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq83_HTML.gif. Note that the G k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq83_HTML.gif’s can evolve in time due to synaptic plasticity mechanisms (see Section 9.4).

The function α k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq84_HTML.gif (called “alpha” profile [44]) mimics the time course of the synaptic conductance upon the occurrence of the spike. Classical examples are:
α k j ( t ) = e t τ k j H ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ9_HTML.gif
(9)
(exponential profile) or:
α k j ( t ) = t τ k j e t τ k j H ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ10_HTML.gif
(10)

with H the Heaviside function (that mimics causality) and τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq85_HTML.gif is the characteristic decay times of the synaptic response. Since t is a time, the division by τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq85_HTML.gif ensures that α k j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq86_HTML.gif is a dimensionless quantity: this eases the legibility of the subsequent equations on physical grounds (dimensionality of physical quantities).

Contrarily to (9) the synaptic profile (10), with α k j ( 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq87_HTML.gif while α k j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq86_HTML.gif is maximal for t = τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq88_HTML.gif, allows one to smoothly delay the spike action on the post-synaptic neuron. More general forms of synaptic responses could be considered as well. For example, the α profile may obey a Green equation of type [45]:
l = 0 k a k j ( l ) d l α k j d u l ( t ) = δ ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equd_HTML.gif

where k = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq89_HTML.gif a k j ( 0 ) = 1 τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq90_HTML.gif a k j ( 1 ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq91_HTML.gif, corresponds to (9), and so on.

3.3 Mathematical constraints on the synaptic responses

In all the paper, we assume that the α k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq84_HTML.gif’s are positive and bounded. Moreover, we assume that:
α k j ( t ) t d τ k j d e t τ k j = def f k j ( t ) , t + , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ11_HTML.gif
(11)

for some integer d. So that α k j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq86_HTML.gif decays exponentially fast as t + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq92_HTML.gif, with a characteristic time τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq85_HTML.gif, the decay time of the evoked post-synaptic potential. This constraint matches all synaptic response kernels that we know (where typically d = 0 , 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq93_HTML.gif) [44, 45].

This has the following consequence. For all t, M < t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq94_HTML.gif integer, r integer, we have, setting t = { t } + [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq95_HTML.gif, where { t } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq96_HTML.gif is the fractional part:
r < M α k j ( t r ) = r < M α k j ( [ t ] r + { t } ) = n > [ t ] M α k j ( n + { t } ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Eque_HTML.gif
Therefore, as M http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq97_HTML.gif,
r < M α k j ( t r ) n [ t ] M + 1 f k j ( n + { t } ) < t M + f k j ( u ) d u = P d ( t M τ k j ) e t M τ k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equf_HTML.gif

where P d ( ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq98_HTML.gif is a polynomial of degree d.

We introduce the following (Hardy) notation: if a function f ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq99_HTML.gif is bounded from above, as t + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq92_HTML.gif, by a function g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq100_HTML.gif we write: f ( t ) g ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq101_HTML.gif. Using this notation, we have therefore:

Proposition 1
r < M α k j ( t r ) P d ( t M τ k j ) e t M τ k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ12_HTML.gif
(12)

as M http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq97_HTML.gif.

Additionally, the constraint (11) implies that there is some α + < + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq102_HTML.gif such that, for all t, for all k, j:
r < t α k j ( t r ) α + . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ13_HTML.gif
(13)
Indeed, for n 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq103_HTML.gif integer, call A k j ( n ) = sup { α k j ( n + x ) ; x [ 0 , 1 [ } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq104_HTML.gif. Then,
r < t α k j ( t r ) = n = 1 + α k j ( n + { t } ) n = 1 + A k j ( n ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equg_HTML.gif
Due to (11), this series converges (e.g., from Cauchy criterion). We set:
α + = max k j n = 1 + A k j ( n ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equh_HTML.gif

On physical grounds, it implies that the conductance g k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq9_HTML.gif remains bounded, even if each pre-synaptic neuron is firing all the time (see Equation (29) below).

3.4 Synaptic summation

Assume that Equation (8) remains valid for an arbitrary number of pre-synaptic spikes emitted by neuron j within a finite time interval [ s , t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq105_HTML.gif (i.e., neglecting non-linear effects such as the fact that there is a finite amount of neurotransmitter leading to saturation effects). Then, one obtains the following equation for the conductance g k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq78_HTML.gif at time t, upon the arrival of spikes at times t j ( r ) ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq50_HTML.gif in the time interval [ s , t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq105_HTML.gif:
g k j ( t ) = g k j ( s ) + G k j { r ; s t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equi_HTML.gif

The conductance at time s, g k j ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq106_HTML.gif, depends on the neuron j’s activity preceding s. This term is therefore unknown unless one knows exactly the past evolution before s. One way to circumvent this problem is to taking s arbitrary far in the past, i.e., taking s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq107_HTML.gif in order to remove the dependence on initial conditions. This corresponds to the following situation. When one observes a real neural network, the time where the observation starts, say t = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq108_HTML.gif, is usually not the time when the system has begun to exist, s in our notations. Taking s arbitrary far in the past corresponds to assuming that the system has evolved long enough so that it has reached sort of a “permanent regime”, not necessarily stationary, when the observation starts. On phenomenological grounds, it is enough to take −s larger than all characteristic relaxation times in the system (e.g., leak rate and synaptic decay rate). Here, for mathematical purposes, it is easier to take the limit s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq107_HTML.gif.

Since g k j ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq80_HTML.gif depends on the raster plot up to time t, via the spiking times t j ( r ) ( ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq50_HTML.gif, this limit makes only sense when taking it “conditionally” to a prescribed raster plot ω. In other words, one can know the value of the conductances g k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq78_HTML.gif at time t only if the past spike times of the network are known. We write g k j ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq109_HTML.gif from now on to make this dependence explicit.

We set
α k j ( t , ω ) = lim s { r ; s t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) { r ; t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ14_HTML.gif
(14)

with the convention that = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq110_HTML.gif so that α k j ( t , Ω 0 ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq111_HTML.gif (recall that Ω0 is the raster such that no neuron ever fires). The limit (14) exists from (13).

3.5 Noise

We allow, in the definition of the current I k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq73_HTML.gif in Equation (6), the possibility of having a stochastic term corresponding to noise so that:
I k ( t ) = i k ( ext ) ( t ) + σ B ξ k ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ15_HTML.gif
(15)

where i k ( ext ) ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq112_HTML.gif is a deterministic external current and ξ k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq113_HTML.gif a noise term whose amplitude is controlled by σ B > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq114_HTML.gif. The model affords an extension where σ B http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq115_HTML.gif depends on k but this extension is straightforward and we do not develop it here. The noise term can be interpreted as the random variation in the ionic flux of charges crossing the membrane per unit time at the post-synaptic button, upon opening of ionic channels due to the binding of neurotransmitter.

We assume that ξ k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq113_HTML.gif is a white noise, ξ k ( t ) = d B k d t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq116_HTML.gif where d B k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq117_HTML.gif is a Wiener process, so that d B ( t ) = ( d B k ( t ) ) k = 1 N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq118_HTML.gif is a N-dimensional Wiener process. Call P the noise probability distribution and E [ ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq119_HTML.gif the expectation under P. Then, by definition, E [ d B k ( t ) ] = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq120_HTML.gif, k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq121_HTML.gif, t R http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq122_HTML.gif, and E [ d B k ( s ) d B l ( t ) ] = δ k l δ ( t s ) d t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq123_HTML.gif where δ k l = 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq124_HTML.gif if l = k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq125_HTML.gif, l , k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq126_HTML.gif and δ ( t s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq127_HTML.gif is the Dirac distribution.

3.6 Differential equation for the integrate regime of gIF

Summarizing, we write Equation (6) in the form:
C k d V k d t + g k ( t , ω ) V k = i k ( t , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ16_HTML.gif
(16)
where:
g k ( t , ω ) = g L , k + j = 1 N G k j α k j ( t , ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ17_HTML.gif
(17)

This is the more general conductance form considered in this paper.

Moreover,
i k ( t , ω ) = g L , k E L + j = 1 N W k j α k j ( t , ω ) + i k ( ext ) ( t ) + σ B ξ k ( t ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ18_HTML.gif
(18)
where W k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq128_HTML.gif is the synaptic weight:
{ W k j = E + G k j , if j E , W k j = E G k j , if j I . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equj_HTML.gif

These equations hold when the membrane potential is below the threshold (Integrate regime).

Therefore, gIF models constitute rather complex dynamical systems: the vector field (r.h.s) of the differential Equation (16) depends on an auxiliary “variable”, which is the past spike sequence ω [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq63_HTML.gif and to define properly the evolution of V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif from time t to later times one needs to know the spikes arising before t. This is precisely what makes gIF models more interesting than LIF. The definition of conductances introduces long-term memory effects.

IF models implement a reset mechanism on the membrane potential: If neuron k has been reset between s and t, say at time τ, then V k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq2_HTML.gif depends only on V k ( τ ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq129_HTML.gif and not on previous values, as in (4). But, in gIF model, contrarily to LIF, there is also a dependence in the past via the conductance and this dependence is not erased by the reset. That is why we have to consider a system with infinite memory.

3.7 The parameters space

The stochastic dynamical system (16) depends on a huge set of parameters: the membrane capacities C k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq6_HTML.gif, k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq130_HTML.gif, the threshold θ, the reversal potentials E L http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq131_HTML.gif, E + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq68_HTML.gif, E http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq70_HTML.gif, the leak conductance g L http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq132_HTML.gif; the maximal synaptic conductances G k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq83_HTML.gif, k , j = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq133_HTML.gif which define the neural network topology; the characteristic times τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq85_HTML.gif, k , j = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq133_HTML.gif of synaptic responses decay; the noise amplitude σ B http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq115_HTML.gif; and additionally, the parameters defining the external current i k ( ext ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq134_HTML.gif. Although some parameters can be fixed from biology, such as C k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq6_HTML.gif, the reversal potentials, τ k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq85_HTML.gif, …some others such as the G k j http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq83_HTML.gif’s must be allowed to vary freely in order to leave open the possibility of modeling very different neural networks structures.

In this paper, we are not interested in describing properties arising for specific values of those parameters, but instead in generic properties that hold on sets of parameters. More specifically, we denote the list of all parameters ( ( C k ) k = 1 N , E L , E + , E , ( G k j ) k , j = 1 N , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq135_HTML.gif by the symbol γ. This is a vector in R K http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq136_HTML.gif where K is the total number of parameters. In this paper, we assume that γ belongs to a bounded subset H R K http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq137_HTML.gif. Basically, we want to avoid situations where some parameters become infinite, which would be unphysical. So the limits of H http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq138_HTML.gif are the limits imposed by biophysics. Additionally, we assume that σ R > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq139_HTML.gif and σ B > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq114_HTML.gif. Together with physical constraints such as “conductances are positive”, these are the only assumption made in parameters. All mathematical results stated in the paper hold for any γ H http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq140_HTML.gif.

4 gIF model dynamics for a fixed raster

We assume that the raster ω is fixed, namely the spike history is given. Then, it is possible to integrate the Equation (16) (Integrate regime) and to obtain explicitly the value of the membrane potential of a neuron at time t, given the membrane potential value at time s. Additionally, the reset condition (4) has the consequence of removing the dependence of neuron k on the past anterior to τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif.

4.1 Integrate regime

For t 1 t 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq141_HTML.gif, t 1 , t 2 R http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq142_HTML.gif, set:
Γ k ( t 1 , t 2 , ω ) = e 1 C k t 1 t 2 g k ( u , ω ) d u . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ19_HTML.gif
(19)
We have:
Γ k ( t 1 , t 1 , ω ) = Γ k ( t 2 , t 2 , ω ) = 1 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equk_HTML.gif
and:
Γ k ( t 1 , t 2 , ω ) t 1 = g k ( t 1 , ω ) C k Γ k ( t 1 , t 2 , ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equl_HTML.gif
Fix two times s < t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq143_HTML.gif and assume that for neuron k, V k ( u ) < θ http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq144_HTML.gif, s u t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq145_HTML.gif so that the membrane potential V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif obeys (16). Then,
t 1 [ Γ k ( t 1 , t 2 , ω ) V k ( t 1 ) ] = Γ k ( t 1 , t 2 , ω ) [ d V k d t 1 + g k ( t 1 , ω ) C k V k ( t 1 ) ] = Γ k ( t 1 , t 2 , ω ) i k ( t 1 , ω ) C k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equm_HTML.gif
We have then integrating the previous equation with respect to t 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq16_HTML.gif between s and t and setting t 2 = t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq146_HTML.gif:
V k ( t ) = Γ k ( s , t , ω ) V k ( s ) + 1 C k s t Γ k ( t 1 , t , ω ) i k ( t 1 , ω ) d t 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equn_HTML.gif

This equation gives the variation in membrane potential during a period of rest (no spike) of the neuron. Note, however, that this neuron can still receive spikes from the other neurons via the update of conductances (made explicit in the previous equation by the dependence in the raster plot ω).

The term Γ k ( s , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq147_HTML.gif given by (19) is an effective leak between s , t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq148_HTML.gif. In the leaky integrate and fire model, it would have been equal to e s t 1 τ L d t 1 = e t s τ L http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq149_HTML.gif. The term:
V k ( s , t , ω ) = def 1 C k s t Γ k ( t 1 , t , ω ) i k ( t 1 , ω ) d t 1 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equo_HTML.gif
has the dimension of a voltage. It corresponds to the integration of the total current between s and t weighted by the effective leak term Γ k ( t 1 , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq150_HTML.gif. It decomposes as
V k ( s , t , ω ) = V k ( syn ) ( s , t , ω ) + V k ( ext ) ( s , t , ω ) + V k ( B ) ( s , t , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equp_HTML.gif
where,
V k ( syn ) ( s , t , ω ) = 1 C k j = 1 N W k j s t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ20_HTML.gif
(20)
is the synaptic contribution. Moreover,
V k ( ext ) ( s , t , ω ) = E L τ L , k s t Γ k ( t 1 , t , ω ) d t 1 + 1 C k s t i k ( ext ) ( t 1 ) Γ k ( t 1 , t , ω ) d t 1 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equq_HTML.gif
where we set:
τ L , k = def C k g L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ21_HTML.gif
(21)

the characteristic leak time of neuron k. We have included the leak reversal potential term in this “external” term for convenience. Therefore, even if there is no external current, this term is nevertheless non-zero.

The sum of the synaptic and external terms gives the deterministic contribution in the membrane potential. We note:
V k ( det ) ( s , t , ω ) = V k ( syn ) ( s , t , ω ) + V k ( ext ) ( s , t , ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equr_HTML.gif
Finally,
V k ( B ) ( s , t , ω ) = def σ B C k s t Γ k ( t 1 , t , ω ) ξ k ( t 1 ) d t 1 = σ B C k s t Γ k ( t 1 , t , ω ) d B k ( t 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ22_HTML.gif
(22)
is a noise term. This is a Gaussian process with mean 0 and variance:
( σ B C k ) 2 E [ ( s t Γ k ( t 1 , t , ω ) d B k ( t 1 ) ) 2 ] = ( σ B C k ) 2 s t Γ k 2 ( t 1 , t , ω ) d t 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ23_HTML.gif
(23)

The square root of this quantity has the dimension of a voltage.

As a final result, for a fixed ω, the variation in membrane potential during a period of rest (no spike) of neuron k between s and t reads (subthreshold oscillations):
V k ( t ) = Γ k ( s , t , ω ) V k ( s ) + V k ( det ) ( s , t , ω ) + V k ( B ) ( s , t , ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ24_HTML.gif
(24)

4.2 Reset

In Equation (4), as in all IF models that we know, the reset of the membrane potential has the effect of removing the dependence of V k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq8_HTML.gif on its past since V k ( [ t + τ sep ] ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq151_HTML.gif is replaced by V reset http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq27_HTML.gif. Hence, reset removes the dependence in the initial condition V k ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq152_HTML.gif in (24) provided that neuron k fires between s and t in the raster ω. As a consequence, Equation (24) holds, from the “last reset time” introduced in Section 2.10 up to time t. Then, Equation (24) reads
V k ( t ) = V k ( det ) ( τ k ( t , ω ) , t , ω ) + V k ( noise ) ( τ k ( t , ω ) , t , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ25_HTML.gif
(25)
where:
V k ( noise ) ( τ k ( t , ω ) , t , ω ) = Γ k ( τ k ( t , ω ) , t , ω ) V reset + V k ( B ) ( τ k ( t , ω ) , t , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ26_HTML.gif
(26)
is a Gaussian process with mean zero and variance:
σ k 2 ( τ k ( t , ω ) , t , ω ) = Γ k 2 ( τ k ( t , ω ) , t , ω ) σ R 2 + ( σ B C k ) 2 τ k ( t , ω ) t Γ k 2 ( t 1 , t , ω ) d t 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ27_HTML.gif
(27)

5 Useful bounds

We now prove several bounds used throughout the paper.

5.1 Bounds on the conductance

From (13), and since α k j ( t ) 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq153_HTML.gif:
0 = α k j ( t , Ω 0 ) α k j ( t , ω ) = { r ; t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) r < t α k j ( t r ) = α k j ( t , Ω 1 ) α + . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ28_HTML.gif
(28)
Therefore,
g L , k = g k ( t , Ω 0 ) g k ( t , ω ) g k ( t , Ω 1 ) = def g M , k g L , k + α + j = 1 N G k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ29_HTML.gif
(29)
so that the conductance is uniformly bounded in t and ω. The minimal conductance is attained when no neuron fires ever so that Ω0 is the “lowest conductance state”. On the opposite, the maximal conductance is reached when all neurons fire all the time so that Ω1 is the “highest conductance state”. To simplify notations, we note τ M , k = C k g M , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq154_HTML.gif. This is the minimal relaxation time scale for neuron k while τ L , k = C k g L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq155_HTML.gif is the maximal relaxation time.
τ M , k = C k g M , k τ L , k = C k g L , k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ30_HTML.gif
(30)

5.2 Bounds on membrane potential

Now, from (19), we have, for s < t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq143_HTML.gif:
0 Γ k ( s , t , Ω 1 ) = e t s τ M , k Γ k ( s , t , ω ) Γ k ( s , t , Ω 0 ) = e t s τ L , k < 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ31_HTML.gif
(31)

As a consequence, Γ k ( s , t , ω ) 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq156_HTML.gif exponentially fast as s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq107_HTML.gif.

Moreover,
0 s t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 α + s t e t t 1 τ L , k d t 1 = α + τ L , k ( 1 e t s τ L , k ) α + τ L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ32_HTML.gif
(32)
so that:
α + g L , k j I W k j V k ( syn ) ( s , t , ω ) α + g L , k j E W k j . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ33_HTML.gif
(33)

Thus, V k ( syn ) ( s , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq157_HTML.gif is uniformly bounded in s , t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq148_HTML.gif.

Establishing similar bounds for V k ( ext ) ( s , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq158_HTML.gif requires the assumption that A i k ( ext ) ( t ) B http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq159_HTML.gif, but obtaining tighter bounds requires additionally the knowledge of the sign of A C k + E L τ L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq160_HTML.gif and of B C k + E L τ L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq161_HTML.gif. Here, we have only to consider that:
| i k ( ext ) ( t ) | i + . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equs_HTML.gif
In this case,
| V k ( ext ) ( s , t , ω ) | = | E L τ L , k s t Γ k ( t 1 , t , ω ) d t 1 + 1 C k s t i k ( ext ) ( t 1 ) Γ k ( t 1 , t , ω ) d t 1 | [ | E L | τ L , k + i + C k ] s t Γ k ( t 1 , t , ω ) d t 1 [ | E L | τ L , k + i + C k ] s t e t t 1 τ L , k d t 1 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equt_HTML.gif
so that:
| V k ( ext ) ( s , t , ω ) | ( | E L | + i + g L , k ) ( 1 e t s τ L , k ) | E L | + i + g L , k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ34_HTML.gif
(34)

Consequently,

Proposition 2
V k = def α + g L , k j I W k j ( | E L | + i + g L , k ) < V k ( det ) ( s , t , ω ) < V k + = def α + g L , k j E W k j + | E L | + i + g L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ35_HTML.gif
(35)

which provides uniform bounds in s, t, ω for the deterministic part of the membrane potential.

5.3 Bounds on the noise variance

Let us now consider the stochastic part V k ( noise ) ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq162_HTML.gif. It has zero mean, and its variance (27) obeys the bounds:
e 2 t τ k ( t , ω ) τ M , k σ R 2 + τ M , k 2 ( σ B C k ) 2 ( 1 e 2 t τ k ( t , ω ) τ M , k ) σ k 2 ( τ k ( t , ω ) , t , ω ) e 2 t τ k ( t , ω ) τ L , k σ R 2 + τ L , k 2 ( σ B C k ) 2 ( 1 e 2 t τ k ( t , ω ) τ L , k ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equu_HTML.gif
If σ R 2 < τ M , k 2 ( σ B C k ) 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq163_HTML.gif the left-hand side is an increasing function of u = t τ k ( t , ω ) 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq164_HTML.gif so that the minimum, σ R 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq32_HTML.gif is reached for u = 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq165_HTML.gif while the maximum is reached for u = + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq166_HTML.gif and is τ M , k 2 ( σ B C k ) 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq167_HTML.gif. The opposite holds if σ R 2 τ M , k 2 ( σ B C k ) 2 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq168_HTML.gif. The same argument holds mutatis mutandis for the right-hand side. We set:
σ k = def min ( σ B C k τ M , k 2 , σ R ) ; σ k + = def max ( σ B C k τ L , k 2 , σ R ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ36_HTML.gif
(36)

so that:

Proposition 3
0 < σ k σ k ( τ k ( t , ω ) , t , ω ) σ k + < + . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ37_HTML.gif
(37)

5.4 The limit τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq169_HTML.gif

For fixed s and t, there are infinitely many rasters such that τ k ( t , ω ) < s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq170_HTML.gif (we remind that rasters are infinite sequences). One may argue that taking the difference t s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq171_HTML.gif sufficiently large, the probability of such sequences should vanish. It is indeed possible to show (Section 8.1) that this probability vanishes exponentially fast with t s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq171_HTML.gif, meaning unfortunately that it is positive whatever t s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq171_HTML.gif. So we have to consider cases where τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq65_HTML.gif can go arbitrary far in past (this is also a key toward an extension of the present analysis to more general conductance-based models as discussed in Section 9.3). Therefore, we have to check that the quantities introduced in the previous sections are well defined as τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq169_HTML.gif.

Fix s real. For all ω such that τ k ( t , ω ) s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq172_HTML.gif - this condition ensuring that k does not fire between s and t - we have, from (28), (31), 0 Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) α + e t t 1 τ L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq173_HTML.gif, t 1 [ s , t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq174_HTML.gif. Now, since lim s s t e t t 1 τ L , k d t 1 = τ L , k http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq175_HTML.gif exists, the limit
lim τ k ( t , ω ) V k ( syn ) ( τ k ( t , ω ) , t , ω ) = 1 C k j = 1 N W k j t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equv_HTML.gif

exists as well. The same holds for the external term V k ( ext ) ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq176_HTML.gif.

Finally, since Γ k ( τ k ( t , ω ) , t , ω ) 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq177_HTML.gif as τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq169_HTML.gif the noise term (26) becomes in the limit:
lim τ k ( t , ω ) V k ( noise ) ( τ k ( t , ω ) , t , ω ) = σ B C k t Γ k ( t 1 , t , ω ) d B k ( t 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equw_HTML.gif

which is a Gaussian process with mean 0 and a variance ( σ B C k ) 2 t Γ k 2 ( t 1 , t , ω ) d t 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq178_HTML.gif which obeys the bounds (37).

6 Continuity with respect to a raster

6.1 Definition

Due to the particular structure of gIF models, we have seen that the membrane potential at time t is both a function of t and of the full sequence of past spikes ω [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq63_HTML.gif. One expects, however, the dependence with respect to the past spikes to decay as those spikes are more distant in the past. This issue is related to a notion of continuity with respect to a raster that we now characterize.

Definition 1 Let m be a positive integer. The m-variation of a function f ( t , ω ) f ( t , ω [ t ] ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq179_HTML.gif is:
var m [ f ( t , ) ] = sup { | f ( t , ω ) f ( t , ω ) | : ω = m , [ t ] ω } . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ38_HTML.gif
(38)

where the definition of = m , [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq180_HTML.gif is given in Equation (5). Hence, this notion characterizes the maximal variation of f ( t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq181_HTML.gif on the set of spikes identical from time [ t ] m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq182_HTML.gif to time [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq21_HTML.gif (cylinder set). It implements the fact that one may truncate the spike history to time [ t ] m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq182_HTML.gif and make an error which is at most var m [ f ( t , ) ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq183_HTML.gif.

Definition 2 The function f ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq61_HTML.gif is continuous if var m [ f ( t , ) ] 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq184_HTML.gif as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq185_HTML.gif.

An additional information is provided by the convergence rate to 0 with m. The faster this convergence, the smaller the error made when replacing an infinite raster by a spike block on a finite time horizon.

6.2 Continuity of conductances

Proposition 4 The conductance g k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq186_HTML.gifis continuous in ω, for all t, for all k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq130_HTML.gif.

Proof Fix k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq130_HTML.gif, t R http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq187_HTML.gif, m > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq188_HTML.gif integer. We have, for ω = m , [ t ] ω http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq189_HTML.gif:
| α k j ( t , ω ) α k j ( t , ω ) | = | { r ; t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) { r ; t j ( r ) ( ω ) < t } α k j ( t t j ( r ) ( ω ) ) | = | { r ; t j ( r ) ( ω ) < t m } α k j ( t t j ( r ) ( ω ) ) { r ; t j ( r ) ( ω ) < t m } α k j ( t t j ( r ) ( ω ) ) | , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equx_HTML.gif
since the set of firing times { t m t j ( r ) ( ω ) < t } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq190_HTML.gif, { t m t j ( n ) ( ω ) < t } http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq191_HTML.gif are identical by hypothesis. So, since α k j ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq192_HTML.gif,
| α k j ( t , ω ) α k j ( t , ω ) | { r ; t j ( r ) ( ω ) < t m } α k j ( t t j ( r ) ( ω ) ) + { r ; t j ( r ) ( ω ) < t m } α k j ( t t j ( r ) ( ω ) ) 2 r < t m α k j ( t r ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equy_HTML.gif
Therefore, as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq193_HTML.gif, from (12) and setting M = t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq194_HTML.gif,
var m [ α k j ( t , ) ] 2 r < M α k j ( t r ) 2 P d ( m τ k j ) e m τ k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equz_HTML.gif

which converges to 0 as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq195_HTML.gif.

Therefore, from (17), g k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq186_HTML.gif is continuous with a variation
var m [ g k ( t , ) ] 2 j = 1 N G k j P d ( m τ k j ) e m τ k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaa_HTML.gif

which converges exponentially fast to 0 as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq195_HTML.gif. □

6.3 Continuity of the membrane potentials

Proposition 5 The deterministic part of the membrane potential, V k ( det ) ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq196_HTML.gif, is continuous and its m-variation decays exponentially fast with m.

Proof In the proof, we shall establish precise upper bounds for the variation in V k ( syn ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq197_HTML.gif, V k ( ext ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq198_HTML.gif since they are used later on for the proof of uniqueness of a Gibbs measure (Section 7.4.1). From the previous result, it is easy to show that, for all ω = m , [ t ] ω http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq189_HTML.gif, t 1 t 2 t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq199_HTML.gif:
| Γ k ( t 1 , t 2 , ω ) Γ k ( t 1 , t 2 , ω ) | Γ k ( t 1 , t 2 , ω ) ( e var m [ g k ] C k ( t 2 t 1 ) 1 ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equab_HTML.gif
Therefore, from (31),
var m [ Γ k ( t 1 , t 2 , ) ] e t 2 t 1 τ L , k ( e var m [ g k ] C k ( t 2 t 1 ) 1 ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equac_HTML.gif

and Γ k ( t 1 , t 2 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq200_HTML.gif is continuous in ω.

Now, the product Γ k ( t 1 , t 2 , ω ) α k j ( t 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq201_HTML.gif is continuous as a product of continuous functions. Moreover,
var m [ Γ k ( t 1 , t 2 , ) α k j ( t 1 , ) ] sup ω X Γ k ( t 1 , t 2 , ω ) var m [ α k j ( t 1 , ) ] + sup ω X α k j ( t 1 , ω ) var m [ Γ k ( t 1 , t 2 , ) ] = e t 2 t 1 τ L , k [ var m [ α k j ( t 1 , ) ] + ( e var m [ g k ] C k ( t 2 t 1 ) 1 ) α k j ( t 1 , Ω 1 ) ] , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equad_HTML.gif
so that:
var m [ Γ k ( t 1 , t 2 , ) α k j ( t 1 , ) ] < e t 2 t 1 τ L , k [ 2 P d ( m τ k j ) e m τ k j + ( e var m [ g k ] C k ( t 2 t 1 ) 1 ) α + ] . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equae_HTML.gif
Since, as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq195_HTML.gif:
e var m [ g k ] C k ( t 2 t 1 ) 1 var m [ g k ] C k ( t 2 t 1 ) 2 ( t 2 t 1 ) C k j = 1 N G k j P d ( m τ k j ) e m τ k j , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaf_HTML.gif
we have,
var m [ Γ k ( t 1 , t 2 , ) α k j ( t 1 , ) ] 2 e t 2 t 1 τ L , k [ P d ( m τ k j ) e m τ k j + α + ( t 2 t 1 ) C k j = 1 N G k j P d ( m τ k j ) e m τ k j ] , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equag_HTML.gif

which converges to 0 as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq195_HTML.gif.

Let us show the continuity of V k ( syn ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq202_HTML.gif. We have, from (20),
| V k ( syn ) ( τ k ( t , ω ) , t , ω ) V k ( syn ) ( τ k ( t , ω ) , t , ω ) | 1 C k j = 1 N | W k j | × | τ k ( t , ω ) t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 τ k ( t , ω ) t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 | . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equah_HTML.gif
The following inequality is used at several places in the paper. For a t 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq16_HTML.gif-integrable function f ( t 1 , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq203_HTML.gif, we have:
| τ k ( t , ω ) t f ( t 1 , t , ω ) d t 1 τ k ( t , ω ) t f ( t 1 , t , ω ) d t 1 | τ k ( t , ω ) t | f ( t 1 , t , ω ) f ( t 1 , t , ω ) | d t 1 + | τ k ( t , ω ) τ k ( t , ω ) f ( t 1 , t , ω ) d t 1 | . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ39_HTML.gif
(39)
Here, it gives, for t 1 t http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq204_HTML.gif:
| τ k ( t , ω ) t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 τ k ( t , ω ) t Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 | τ k ( t , ω ) t | Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) | d t 1 + | τ k ( t , ω ) τ k ( t , ω ) Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 | . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equai_HTML.gif
For the first term, we have,
τ k ( t , ω ) t | Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) | d t 1 τ k ( t , ω ) t var m [ Γ k ( t 1 , t , ) α k j ( t 1 , ) ] d t 1 t var m [ Γ k ( t 1 , t , ) α k j ( t 1 , ) ] d t 1 t 2 e t t 1 τ L , k [ P d ( m τ k j ) e m τ k j + α + ( t t 1 ) C k j = 1 N G k j P d ( m τ k j ) e m τ k j ] d t 1 = 2 P d ( m τ k j ) e m τ k j t e t t 1 τ L , k d t 1 + 2 α + C k j = 1 N G k j P d ( m τ k j ) e m τ k j t ( t t 1 ) e t t 1 τ L , k d t 1 = 2 τ L , k [ P d ( m τ k j ) e m τ k j + α + τ L , k C k j = 1 N G k j P d ( m τ k j ) e m τ k j ] . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaj_HTML.gif
Let us now consider the second term. If τ k ( t , ω ) t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq205_HTML.gif or τ k ( t , ω ) t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq206_HTML.gif, then τ k ( t , ω ) = τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq207_HTML.gif and this term vanishes. Therefore, the supremum in the definition of var m [ V k ( syn ) ( τ k ( t , ) , t , ) ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq208_HTML.gif is attained if τ k ( t , ω ) < t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq209_HTML.gif and τ k ( t , ω ) < t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq210_HTML.gif. We may assume, without loss of generality, that τ k ( t , ω ) τ k ( t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq211_HTML.gif. Then, from (32),
τ k ( t , ω ) τ k ( t , ω ) Γ k ( t 1 , t , ω ) α k j ( t 1 , ω ) d t 1 < α + τ k ( t , ω ) τ k ( t , ω ) e t t 1 τ L , k d t 1 = α + τ L , k e t τ k ( t , ω ) τ L , k ( 1 e τ k ( t , ω ) τ k ( t , ω ) τ L , k ) α + τ L , k e t τ k ( t , ω ) τ L , k α + τ L , k e m τ L , k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equak_HTML.gif
So, we have, for the variation of V k ( syn ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq202_HTML.gif, using (21):
var m [ V k ( syn ) ( τ k ( t , ) , t , ) ] 1 g L , k j = 1 N | W k j | [ 2 ( P d ( m τ k j ) e m τ k j + α + g L , k j = 1 N G k j P d ( m τ k j ) e m τ k j ) 1 g L , k j = 1 N | W k j | [ + α + e m τ L , k ] , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equal_HTML.gif
so that finally,
var m [ V k ( syn ) ( τ k ( t , ) , t , ) ] j = 1 N A k j ( syn ) P d ( m τ k j ) e m τ k j + B k ( syn ) e m τ L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ40_HTML.gif
(40)
with
A k j ( syn ) = 1 g L , k ( 2 | W k j | + α + G k j g L , k j = 1 N | W k j | ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ41_HTML.gif
(41)
B k ( syn ) = α + g L , k j = 1 N | W k j | , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ42_HTML.gif
(42)

and var m [ V k ( syn ) ( τ k ( t , ) , t , ) ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq212_HTML.gif converges to 0 exponentially fast as m + http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq195_HTML.gif.

Now, let us show the continuity of V k ( ext ) ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq213_HTML.gif with respect to ω. We have:
| V k ( ext ) ( τ k ( t , ω ) , t , ω ) V k ( ext ) ( τ k ( t , ω ) , t , ω ) | = | τ k ( t , ω ) t ( E L τ L , k + i k ( ext ) ( t 1 ) C k ) Γ k ( t 1 , t , ω ) d t 1 | τ k ( t , ω ) t ( E L τ L , k + i k ( ext ) ( t 1 ) C k ) Γ k ( t 1 , t , ω ) d t 1 | τ k ( t , ω ) t | E L τ L , k + i k ( ext ) ( t 1 ) C k | | Γ k ( t 1 , t , ω ) Γ k ( t 1 , t , ω ) | d t 1 + τ k ( t , ω ) τ k ( t , ω ) | E L τ L , k + i k ( ext ) ( t 1 ) C k | Γ k ( t 1 , t , ω ) d t 1 ( | E L | τ L , k + i + C k ) × ( τ k ( t , ω ) t | Γ k ( t 1 , t , ω ) Γ k ( t 1 , t , ω ) | d t 1 + τ k ( t , ω ) τ k ( t , ω ) Γ k ( t 1 , t , ω ) d t 1 ) ( | E L | τ L , k + i + C k ) ( τ k ( t , ω ) t var m [ Γ k ( t 1 , t , ) ] d t 1 + τ k ( t , ω ) τ k ( t , ω ) e t t 1 τ L , k d t 1 ) ( | E L | τ L , k + i + C k ) ( 2 τ L , k 2 C k j = 1 N G k j P d ( m τ k j ) e m τ k j + τ L , k e m τ L , k ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equam_HTML.gif
where, in the last inequality, we have used that the supremum in the variation is attained for τ k ( t , ω ) < t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq209_HTML.gif and τ k ( t , ω ) < t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq210_HTML.gif. Finally:
var m [ V k ( ext ) ( τ k ( t , ) , t , ) ] j = 1 N A k j ( ext ) P d ( m τ k j ) e m τ k j + B k ( ext ) e m τ L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ43_HTML.gif
(43)
where,
A k j ( ext ) = 2 G k j g L , k B k ( ext ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ44_HTML.gif
(44)
B k ( ext ) = | E L | + i + g L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ45_HTML.gif
(45)

and V k ( ext ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq214_HTML.gif is continuous.

As a conclusion, V k ( det ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq215_HTML.gif is continuous as the sum of two continuous functions. □

6.4 Continuity of the variance of V k ( noise ) ( τ k ( t , ) , t , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq216_HTML.gif

Using the same type of arguments, one can also prove that

Proposition 6 The variance σ k ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq217_HTML.gifis continuous in ω, for all t, for all k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq130_HTML.gif.

Proof We have, from (27)
| σ k 2 ( τ k ( t , ω ) , t , ω ) σ k 2 ( τ k ( t , ω ) , t , ω ) | σ R 2 var m [ Γ k 2 ( τ k ( t , ) , t , ) ] + ( σ B C k ) 2 | τ k ( t , ω ) t Γ k 2 ( t 1 , t , ω ) d t 1 τ k ( t , ω ) t Γ k 2 ( t 1 , t , ω ) d t 1 | . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equan_HTML.gif
For the first term, we have that the sup in var m [ Γ k 2 ( τ k ( t , ) , t , ) ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq218_HTML.gif is attained for τ k ( t , ω ) , τ k ( t , ω ) < t m http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq219_HTML.gif and:
var m [ Γ k 2 ( τ k ( t , ) , t , ) ] 2 sup { Γ k 2 ( τ k ( t , ω ) , t , ω ) ; ω s.t. τ k ( t , ω ) < t m } 2 e 2 m τ L , k . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equao_HTML.gif
For the second term, we have:
| τ k ( t , ω ) t Γ k 2 ( t 1 , t , ω ) d t 1 τ k ( t , ω ) t Γ k 2 ( t 1 , t , ω ) d t 1 | τ k ( t , ω ) t | Γ k 2 ( t 1 , t , ω ) Γ k 2 ( t 1 , t , ω ) | d t 1 + τ k ( t , ω ) τ k ( t , ω ) Γ k 2 ( t 1 , t , ω ) d t 1 τ k ( t , ω ) t var m [ Γ k 2 ( t 1 , t , ) ] d t 1 + τ k ( t , ω ) τ k ( t , ω ) e 2 ( t t 1 ) τ L , k d t 1 t e 2 ( t t 1 ) τ L , k ( e 2 var m [ g k ( t , ) ( t t 1 ) ] C k 1 ) d t 1 + τ L , k 2 e 2 m τ L , k τ L , k 2 2 C k var m [ g k ( t , ) ] + τ L , k 2 e 2 m τ L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equap_HTML.gif
so that finally,
var m [ σ k 2 ( t , ) ] j = 1 N A k j ( σ ) P d ( m τ k j ) e m τ k j + C k ( σ ) e 2 m τ L , k , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ46_HTML.gif
(46)
with
A k j ( σ ) = G k j g L , k ( σ B τ L , k C k ) 2 , C k ( σ ) = 1 2 ( σ B τ L , k C k ) 2 + 2 σ R 2 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaq_HTML.gif

and continuity follows. □

6.5 Remark

Note that the variation in all quantities considered here is exponentially decaying with a time constant given by max ( τ k j , τ L , k ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq220_HTML.gif. This is physically satisfactory: the loss of memory in the system is controlled by the leak time and the decay of the post-synaptic potential.

7 Statistics of raster plots

7.1 Conditional probability distribution of V k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq2_HTML.gif

Recall that P is the joint distribution of the noise and E [ ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq119_HTML.gif the expectation under P. Under P, the membrane potential V is a stochastic process whose evolution, below the threshold, is given Equations (24), (25) and above by (4). It follows from the previous analysis that:

Proposition 7 Conditionally to ω [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq63_HTML.gif, V ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq3_HTML.gifis Gaussian with mean:
E [ V k ( t ) | ω [ t ] ] = V k ( det ) ( τ k ( t , ω ) , t , ω ) , k = 1 , , N , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equar_HTML.gif
and covariance:
Cov [ V k ( t ) , V l ( t ) | ω [ t ] ] = σ k 2 ( τ k ( t , ω ) , t , ω ) δ k l , k , l = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equas_HTML.gif

where σ k 2 ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq221_HTML.gifis given by (27).

Moreover, the V k ( t ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq2_HTML.gif’s, k = 1 , , N http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq130_HTML.gifare conditionally independent.

Proof Essentially, the proof is a direct consequence of Equations (24), (25) and the Gaussian nature of the noise V k ( noise ) ( τ k ( t , ω ) , t , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq222_HTML.gif. The conditional independence results from the fact that:
Cov [ V k ( t ) , V l ( t ) | ω [ t ] ] = σ B 2 C k C l E [ τ k ( t , ω ) t Γ k ( t 1 , t , ω ) d B k ( t 1 ) τ l ( t , ω ) t Γ l ( t 2 , t , ω ) d B l ( t 2 ) | ω ] + Cov [ Γ k ( τ k ( t , ω ) , t , ω ) V reset , Γ l ( τ l ( t , ω ) , t , ω ) V reset ] = σ B 2 C k C l τ k ( t , ω ) t τ l ( t , ω ) t Γ k ( t 1 , t , ω ) Γ l ( t 2 , t , ω ) E [ d B k ( t 1 ) d B l ( t 2 ) ] + σ R 2 Γ k 2 ( τ k ( t , ω ) , t , ω ) δ k l = δ k l [ ( σ B C k ) 2 τ k ( t , ω ) t τ k ( t , ω ) t Γ k ( t 1 , t , ω ) Γ k ( t 2 , t , ω ) δ ( t 1 t 2 ) d t 1 d t 2 δ k l [ + σ R 2 Γ k 2 ( τ k ( t , ω ) , t , ω ) ] = σ k 2 ( τ k ( t , ω ) , t , ω ) δ k l . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equat_HTML.gif

 □

7.2 The transition probability

We now compute the probability of a spiking pattern at time n = [ t ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq223_HTML.gif, ω ( n ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq224_HTML.gif, given the past sequence ω n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq225_HTML.gif.

Proposition 8 The probability of ω ( n ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq224_HTML.gifconditionally to ω n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq225_HTML.gifis given by:

P [ ω ( n ) | ω n 1 ] = k = 1 N P [ ω k ( n ) | ω n 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ47_HTML.gif
(47)
with
P [ ω k ( n ) | ω n 1 ] = ω k ( n ) π ( X k ( n 1 , ω ) ) + ( 1 ω k ( n ) ) ( 1 π ( X k ( n 1 , ω ) ) ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ48_HTML.gif
(48)
where
X k ( n 1 , ω ) = θ V k ( det ) ( τ k ( n 1 , ω ) , n 1 , ω ) σ k ( τ k ( n 1 , ω ) , n 1 , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ49_HTML.gif
(49)
and
π ( x ) = 1 2 π x + e u 2 2 d u . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ50_HTML.gif
(50)
Proof We have, using the conditional independence of the V k ( n ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq226_HTML.gif’s:
P [ ω ( n ) | ω n 1 ] = k = 1 N ( ω k ( n ) P [ V k ( n 1 ) θ | ω n 1 ] + ( 1 ω k ( n ) ) P [ V k ( n 1 ) < θ | ω n 1 ] ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equau_HTML.gif

Since the V k ( n 1 ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq227_HTML.gif’s are conditionally Gaussian, with mean V k ( det ) ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq228_HTML.gif and variance σ k 2 ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq229_HTML.gif, we directly obtain (47), (48).

Note that since σ k ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq230_HTML.gif is bounded from below by a positive quantity (see (37)), the ratio θ V k ( det ) ( τ k ( n 1 , ω ) , n 1 , ω ) σ k ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq231_HTML.gif in (48) is defined for all ω X http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq47_HTML.gif. □

7.3 Chains with complete connections

The transition probabilities (47) define a stochastic process on the set of raster plots where the underlying membrane potential dynamics is summarized in the terms V k ( det ) ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq232_HTML.gif and σ k ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq230_HTML.gif. While the integral defining these terms extends from τ k ( n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq233_HTML.gif to n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq234_HTML.gif where τ k ( n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq233_HTML.gif can go arbitrary far in the past, the integrand involves the conductance g k ( n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq235_HTML.gif that summarizes an history dating back to s = http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq236_HTML.gif. As a consequence, the probability transitions generate a stochastic process with unbounded memory, thus non-Markovian. One may argue that this property is a result of our procedure of taking the initial condition in a infinite past s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq107_HTML.gif, to remove the unresolved dependency on g k ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq237_HTML.gif (Section 3.4). So the alternative is either to keep s finite in order to have a Markovian process; then, we have to fix arbitrarily g k ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq237_HTML.gif and the probability distribution of V k ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq152_HTML.gif. Or we take s http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq107_HTML.gif, removing the initial condition, to the price of considering a non-Markovian process. Actually, such processes are widely studied in the literature under the name of “chains with complete connections” [43, 4649] and several important results can be used here. So we adopt the second approach of the alternative. As a by-product, the knowledge of the Gibbs measure provided by this analysis allows a posteriori to fix the probability distribution of V k ( s ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq152_HTML.gif.

For the sake of completeness, we give here the definition of a chain with complete connections (see [43] for more details). For n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif, we note A n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq238_HTML.gif the set of sequences ω n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq225_HTML.gif and F n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq239_HTML.gif the related σ-algebra, while F http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq240_HTML.gif is the σ-algebra related with X = A Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq241_HTML.gif. P ( X , F ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq242_HTML.gif is the set of probability measures on ( X , F ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq243_HTML.gif.

Definition 3 A system of transition probabilities is a family { P n } n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq244_HTML.gif of functions
P n [ | ] : A × A n 1 [ 0 , 1 ] , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equav_HTML.gif

such that the following conditions hold for every n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif:

  • For every ω ( n ) A http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq245_HTML.gif, the function P n [ ω ( n ) | . ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq246_HTML.gif is measurable with respect to F n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq239_HTML.gif.

  • For every ω n 1 A n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq247_HTML.gif,
    ω ( n ) A P n [ ω ( n ) | ω n 1 ] = 1 . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaw_HTML.gif
A probability measure μ in P ( X , F ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq242_HTML.gif is consistent with a system of transition probabilities { P n } n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq244_HTML.gif if for all n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif and all F n http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq248_HTML.gif-measurable functions f:
f ( ω n ) μ ( d ω ) = ω ( n ) A f ( ω n 1 ω ( n ) ) P n [ ω ( n ) | ω n 1 ] μ ( d ω ) . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equax_HTML.gif

Such a measure μ is called a “chain with complete connections consistent with the system of transition probabilities { P n } n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq249_HTML.gif”.

The transitions probabilities (47) constitute such a system of transitions probabilities: the summation to 1 is obvious while the measurability follows from the continuity of P n [ ω ( n ) | ω n 1 ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq250_HTML.gif proved below. To simplify notations, we write p ( n , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq251_HTML.gif instead of P n [ ω ( n ) | ω n 1 ] http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq250_HTML.gif whenever it makes no confusion.

7.4 Existence of a consistent probability measure μ

In the definition above, the measure μ summarizes the statistics of spike trains from −∞ to +∞. Its marginals allow the characterization of finite spike blocks. So, μ provides the characterization of spike train statistics in gIF models. Its existence is established by a standard result in the frame of chains with complete connections stating that a system of continuous transition probabilities on a compact space has at least one probability measure consistent with it [43]. Since π is a continuous function the continuity of p ( n , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq251_HTML.gif with respect to ω follows from the continuity of V k ( det ) ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq232_HTML.gif and the continuity of σ k ( τ k ( n 1 , ω ) , n 1 , ω ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq230_HTML.gif, proved in Section 6.

Therefore, there is at least one probability measure consistent with (47).

7.4.1 The Gibbs distribution

A system of transition probabilities is non-null if for all n Z http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq44_HTML.gif and all ω n 1 A n 1 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq247_HTML.gif P [ ω ( n ) | ω n 1 ] > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq252_HTML.gif. Following [50], a chain with complete connection μ is a Gibbs measure consistent with the system of transition probabilities p ( n , ) http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq253_HTML.gif if this system is continuous and non-null. Gibbs distributions play an important role in statistical physics, as well as ergodic theory and stochastic processes. In statistical physics, they are usually derived from the maximal entropy principle [14]. Here, we use them in a more general context affording to consider non-stationary processes. It turns out that the spike train statistics in gIF model is given by such a Gibbs measure. In this section, we prove the main mathematical result of this paper (uniqueness of the Gibbs measure). The consequences for spike trains characterizations are discussed in the next section.

Theorem 1 For each choice of parameters γ H http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq140_HTML.gif, the gIF model (16) has a unique Gibbs distribution.

The proof of uniqueness is based on the following criteria due to Fernandez and Maillard [50].

Proposition 9 Let:2
m ( p ) = inf n Z inf ω A n p ( n , ω ) , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equay_HTML.gif
and
v ( p ) = sup m Z n m var n m [ p ( n , ) ] . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equaz_HTML.gif

If m ( p ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq254_HTML.gifand v ( p ) < http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq255_HTML.gif, then there exists at most one Gibbs measure consistent with it.

So, to prove the uniqueness, we only have to establish that
m ( p ) > 0 , http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ51_HTML.gif
(51)
v ( p ) < + . http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_Equ52_HTML.gif
(52)

Proof m ( p ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F2190-8567-1-8/MediaObjects/13408_2011_Article_7_IEq256_HTML.gif.

Recall that:
p ( n ,