Statistics of spike trains in conductancebased neural networks: Rigorous results
 Bruno Cessac^{1}Email author
DOI: 10.1186/2190856718
© Cessac; licensee Springer 2011
Received: 26 January 2011
Accepted: 25 August 2011
Published: 25 August 2011
Abstract
We consider a conductancebased 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 timedependent stimulus and apply therefore to nonstationary 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 nonlinear 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 [6–9]). 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 (timetranslation 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., [15–17]), although some authors raised solid objections on this model [12, 18–20] while several papers have pointed out the importance of temporal patterns of activity at the network level [21–23]. As a consequence, a few authors [13, 24, 25] have attempted to define timedependent models of Gibbs distributions where constraints include timedependent 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 spatiotemporal 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 timedependent stimulus generating a timedependent 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 nonlinear 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 discretetime version of Leaky IntegrateandFire 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 nonMarkovian (for recent examples of nonMarkovian 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 nuplets 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 timeconstant) 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 conductancebased 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 timedependent stimulus. Moreover, the postsynaptic 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 nonMarkovian 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 Isinglike models. As a byproduct 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 nonconstant 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 discretetime 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 $k\in \{1,\dots ,N\}$, a variable ${V}_{k}(t)$ 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 $V(t)$ the vector ${({V}_{k}(t))}_{k=1}^{N}$.
We focus here on “integrate and fire models”, where dynamics always consists of two regimes.
2.1 The “integrate regime”
where ${C}_{k}$ is the membrane capacity of neuron k. In its most general form, the neuron k’s membrane conductance ${g}_{k}>0$ depends on ${V}_{k}$ plus additional variables such as the probability of having ionic channels open (see, e.g., HodgkinHuxley equations [35]) as well as on time t. The explicit form of ${g}_{k}$ in the present model is developed in Section 3.4. The current ${i}_{k}$ 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
where ${g}_{k}$ is a constant and ${\tau}_{L}=\frac{{C}_{k}}{{g}_{k}}$ is the characteristic time for membrane potential decay when no current is present (“leak term”).
2.3 Spikes
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 ${t}_{1}$ and neuron 2 at time ${t}_{2}$ with ${t}_{2}{t}_{1}<\delta =1$ 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 $\delta \to 0$ 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
For convenience and in order to simplify the notations in the mathematical developments, we call $[t]$ the largest integer which is ≤t (thus $[1.2]=2$ and $[1.2]=1$). Thus, the integer immediately after t is $[t+1]$ and we have therefore that ${\omega}_{k}([t+1])=1$ whenever ${V}_{k}(t)=\theta $. Although, characteristic events in a raster plot are spikes (neuron fires), it is useful in subsequent developments to consider also the case when neuron is not firing (${\omega}_{k}(n)=0$).
2.6 Reset
The reason why the reset time is the integer number $[t+{\tau}_{\mathrm{sep}}]$ instead of the real $t+{\tau}_{\mathrm{sep}}$ 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 ${V}_{\mathrm{reset}}$ 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 ${\sigma}_{R}^{2}>0$. 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 ${V}_{k}$ 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 ${\sigma}_{R}^{2}>0$ is necessary for our mathematical developments (see the bounds (37)). We assume ${\sigma}_{R}^{2}$ to be small to avoid trivial and unrealistic situations where ${V}_{\mathrm{reset}}\ge \theta $ 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 ${V}_{\mathrm{reset}}$ 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 repolarization phase due to the nonlinear effects of gated ionic channels on the conductance. This leads to introduce, in modeling, additional variables such as activation/inactivation probabilities as in the HodgkinHuxley model [35] or adaptation current as, e.g., in FitzHughNagumo model [39–42] (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 ${V}_{k}(t)$ during the time interval $(t,[t+{\tau}_{\mathrm{sep}}])$. 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)=\theta $, 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 $\omega (n)={({\omega}_{k}(n))}_{k=1}^{N}$. For $m<n$, we note ${\omega}_{m}^{n}=\{\omega (m)\omega (m+1)\cdots \omega (n)\}$ the ordered sequence of spiking patterns between m and n. Such sequences are called spike blocks. Additionally, we note that ${\omega}_{m}^{{n}_{1}1}{\omega}_{{n}_{1}}^{n}={\omega}_{m}^{n}$, the concatenation of the blocks ${\omega}_{m}^{{n}_{1}1}$ and ${\omega}_{{n}_{1}}^{n}$.
Call $\mathcal{A}={\{0,1\}}^{N}$ the set of spiking patterns (alphabet). An element of $X\stackrel{\mathrm{def}}{=}{\mathcal{A}}^{\mathbb{Z}}$, i.e., a biinfinite ordered sequence $\omega ={\{\omega (n)\}}_{n=\infty}^{+\infty}$ of spiking patterns, is called a “raster plot”. It tells us which neurons are firing at each time $n\in \mathbb{Z}$. In experiments, raster plots are obviously finite sequences of spiking pattern but the extension to $\mathbb{Z}$, especially the possibility of considering an arbitrary distant past (negative times), is a key of the present work. In particular, the notation ${\omega}_{\infty}^{n}$ refers to spikes occurring from −∞ to n.
To each raster $\omega \in X$ and each neuron index $j=1,\dots ,N$, we associate an ordered (generically infinite) list of “spike times” ${\{{t}_{j}^{(r)}(\omega )\}}_{r=1}^{\infty}$ (integer numbers) such that ${t}_{j}^{(r)}(\omega )$ is the rth time of firing of neuron j in the raster ω. In other words, we have ${\omega}_{j}(n)=1$ if and only if $n={t}_{j}^{(r)}(\omega )$ for some $r=1,\dots ,+\infty $. We use here the following convention. The index k is used for a postsynaptic neuron while the index j refers to presynaptic neurons. Spiking events are used to update the conductance of neuron k according to spikes emitted by presynaptic 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 Ω_{0} the raster such that ${\omega}_{k}(n)=0$, $\forall k=1,\dots ,N$, $\forall n\in \mathbb{Z}$ (no neuron ever fires) and Ω_{1} the raster ${\omega}_{k}(n)=1$, $\forall i=1,\dots ,N$, $\forall n\in \mathbb{Z}$ (each neuron is firing at every integer time).
For simplicity, we consider that ${\tau}_{\mathrm{ref}}$, the refractory period, is smaller than 1 so that a neuron can fire two consecutive time steps (i.e., one can have ${\omega}_{k}(n)=1$ and ${\omega}_{k}(n+1)=1$). This constraint is discussed in Section 9.2.
2.9 Representation of timedependent functions
Throughout the paper, we use the following convention. For a real function of t and ω, we write $f(t,\omega )$ for $f(t,{\omega}_{\infty}^{[t]})$ 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 $f(t,\omega )$ is a function of the continuous variable t and of the spike block ${\omega}_{\infty}^{[t]}$, where by definition $[t]\le t$, namely $f(t,\omega )$ depends on the spike sequences occurring before t. This constraint is imposed by causality.
2.10 Last reset time
We define ${\tau}_{k}(t,\omega )$ 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 ${\tau}_{k}(t,\omega )$ 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 ${V}_{\mathrm{reset}}$ at time ${\tau}_{k}(t,\omega )$ and by the further subthreshold evolution (1) from time ${\tau}_{k}(t,\omega )$ to time t.
3 Generalized integrate and fire models
In this paper, we concentrate on an extension of (2), called “generalized IntegrateandFire” (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 $\mathcal{E}$) or inhibitory (population $\mathcal{I}$). This is modeled by introducing reversal potentials ${E}^{+}$ for excitatory (typically ${E}^{+}\simeq 0\text{mV}$ for AMPA and NMDA) and ${E}^{}$ for inhibitory (${E}^{}\simeq 70\text{mV}$ for GABA A and ${E}^{}\simeq 95\text{mV}$ 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 ${I}_{k}(t)$. We assume that this current has some stochastic component that mimics synaptic noise (Section 3.5).
where ${g}_{kj}$ is the conductance of the synaptic contact $j\to k$.
3.2 Conductance update upon a spike occurrence
In this equation, the quantity ${G}_{kj}\ge 0$ characterizes the maximal amplitude of the conductance during a postsynaptic potential. We use the convention that ${G}_{kj}=0$ 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 ${G}_{kj}$. Note that the ${G}_{kj}$’s can evolve in time due to synaptic plasticity mechanisms (see Section 9.4).
with H the Heaviside function (that mimics causality) and ${\tau}_{kj}$ is the characteristic decay times of the synaptic response. Since t is a time, the division by ${\tau}_{kj}$ ensures that ${\alpha}_{kj}(t)$ is a dimensionless quantity: this eases the legibility of the subsequent equations on physical grounds (dimensionality of physical quantities).
where $k=1$${a}_{kj}^{(0)}=\frac{1}{{\tau}_{kj}}$${a}_{kj}^{(1)}=1$, corresponds to (9), and so on.
3.3 Mathematical constraints on the synaptic responses
for some integer d. So that ${\alpha}_{kj}(t)$ decays exponentially fast as $t\to +\infty $, with a characteristic time ${\tau}_{kj}$, the decay time of the evoked postsynaptic potential. This constraint matches all synaptic response kernels that we know (where typically $d=0,1$) [44, 45].
where ${P}_{d}()$ is a polynomial of degree d.
We introduce the following (Hardy) notation: if a function $f(t)$ is bounded from above, as $t\to +\infty $, by a function $g(t)$ we write: $f(t)\u2aafg(t)$. Using this notation, we have therefore:
as$M\to \infty $.
On physical grounds, it implies that the conductance ${g}_{k}$ remains bounded, even if each presynaptic neuron is firing all the time (see Equation (29) below).
3.4 Synaptic summation
The conductance at time s, ${g}_{kj}(s)$, 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\to \infty $ 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 $t=0$, 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\to \infty $.
Since ${g}_{kj}(t)$ depends on the raster plot up to time t, via the spiking times ${t}_{j}^{(r)}(\omega )$, 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 ${g}_{kj}$ at time t only if the past spike times of the network are known. We write ${g}_{kj}(t,\omega )$ from now on to make this dependence explicit.
with the convention that ${\sum}_{\varnothing}=0$ so that ${\alpha}_{kj}(t,{\Omega}_{0})=0$ (recall that Ω_{0} is the raster such that no neuron ever fires). The limit (14) exists from (13).
3.5 Noise
where ${i}_{k}^{(\mathrm{ext})}(t)$ is a deterministic external current and ${\xi}_{k}(t)$ a noise term whose amplitude is controlled by ${\sigma}_{B}>0$. The model affords an extension where ${\sigma}_{B}$ 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 postsynaptic button, upon opening of ionic channels due to the binding of neurotransmitter.
We assume that ${\xi}_{k}(t)$ is a white noise, ${\xi}_{k}(t)=\frac{\mathrm{d}{B}_{k}}{\mathrm{d}t}$ where $\mathrm{d}{B}_{k}(t)$ is a Wiener process, so that $\mathrm{d}B(t)={(\mathrm{d}{B}_{k}(t))}_{k=1}^{N}$ is a Ndimensional Wiener process. Call P the noise probability distribution and $\mathbb{E}[]$ the expectation under P. Then, by definition, $\mathbb{E}[\mathrm{d}{B}_{k}(t)]=0$, $\forall k=1,\dots ,N$, $t\in \mathbb{R}$, and $\mathbb{E}[\mathrm{d}{B}_{k}(s)\phantom{\rule{0.2em}{0ex}}\mathrm{d}{B}_{l}(t)]={\delta}_{kl}\delta (ts)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t$ where ${\delta}_{kl}=1$ if $l=k$, $l,k=1,\dots ,N$ and $\delta (ts)$ is the Dirac distribution.
3.6 Differential equation for the integrate regime of gIF
This is the more general conductance form considered in this paper.
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 ${\omega}_{\infty}^{[t]}$ and to define properly the evolution of ${V}_{k}$ 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 longterm 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 ${V}_{k}(t)$ depends only on ${V}_{k}(\tau )$ 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 ${C}_{k}$, $k=1,\dots ,N$, the threshold θ, the reversal potentials ${E}_{L}$, ${E}^{+}$, ${E}^{}$, the leak conductance ${g}_{L}$; the maximal synaptic conductances ${G}_{kj}$, $k,j=1,\dots ,N$ which define the neural network topology; the characteristic times ${\tau}_{kj}$, $k,j=1,\dots ,N$ of synaptic responses decay; the noise amplitude ${\sigma}_{B}$; and additionally, the parameters defining the external current ${i}_{k}^{(\mathrm{ext})}$. Although some parameters can be fixed from biology, such as ${C}_{k}$, the reversal potentials, ${\tau}_{kj}$, …some others such as the ${G}_{kj}$’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 $({({C}_{k})}_{k=1}^{N},{E}_{L},{E}^{+},{E}^{},{({G}_{kj})}_{k,j=1}^{N},\dots )$ by the symbol γ. This is a vector in ${\mathbb{R}}^{K}$ where K is the total number of parameters. In this paper, we assume that γ belongs to a bounded subset $\mathcal{H}\subset {\mathbb{R}}^{K}$. Basically, we want to avoid situations where some parameters become infinite, which would be unphysical. So the limits of $\mathcal{H}$ are the limits imposed by biophysics. Additionally, we assume that ${\sigma}_{R}>0$ and ${\sigma}_{B}>0$. 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 $\gamma \in \mathcal{H}$.
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 ${\tau}_{k}(t,\omega )$.
4.1 Integrate regime
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 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 nonzero.
The square root of this quantity has the dimension of a voltage.
4.2 Reset
5 Useful bounds
We now prove several bounds used throughout the paper.
5.1 Bounds on the conductance
5.2 Bounds on membrane potential
As a consequence, ${\Gamma}_{k}(s,t,\omega )\to 0$ exponentially fast as $s\to \infty $.
Thus, ${\mathcal{V}}_{k}^{(\mathrm{syn})}(s,t,\omega )$ is uniformly bounded in $s,t$.
Consequently,
which provides uniform bounds in s, t, ω for the deterministic part of the membrane potential.
5.3 Bounds on the noise variance
so that:
5.4 The limit ${\tau}_{k}(t,\omega )\to \infty $
For fixed s and t, there are infinitely many rasters such that ${\tau}_{k}(t,\omega )<s$ (we remind that rasters are infinite sequences). One may argue that taking the difference $ts$ 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 $ts$, meaning unfortunately that it is positive whatever $ts$. So we have to consider cases where ${\tau}_{k}(t,\omega )$ can go arbitrary far in past (this is also a key toward an extension of the present analysis to more general conductancebased models as discussed in Section 9.3). Therefore, we have to check that the quantities introduced in the previous sections are well defined as ${\tau}_{k}(t,\omega )\to \infty $.
exists as well. The same holds for the external term ${\mathcal{V}}_{k}^{(\mathrm{ext})}({\tau}_{k}(t,\omega ),t,\omega )$.
which is a Gaussian process with mean 0 and a variance ${(\frac{{\sigma}_{B}}{{C}_{k}})}^{2}{\int}_{\infty}^{t}{\Gamma}_{k}^{2}({t}_{1},t,\omega )\phantom{\rule{0.2em}{0ex}}\mathrm{d}{t}_{1}$ 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 ${\omega}_{\infty}^{[t]}$. 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.
where the definition of $\stackrel{\mathrm{m},[\mathrm{t}]}{=}$ is given in Equation (5). Hence, this notion characterizes the maximal variation of $f(t,\cdot )$ on the set of spikes identical from time $[t]m$ to time $[t]$ (cylinder set). It implements the fact that one may truncate the spike history to time $[t]m$ and make an error which is at most ${var}_{m}[f(t,\cdot )]$.
Definition 2 The function $f(t,\omega )$ is continuous if ${var}_{m}[f(t,\cdot )]\to 0$ as $m\to +\infty $.
An additional information is provided by the convergence rate to 0 with m. The faster this convergence, the smaller the error made when replacing an infinite raster by a spike block on a finite time horizon.
6.2 Continuity of conductances
Proposition 4 The conductance${g}_{k}(t,\omega )$is continuous in ω, for all t, for all$k=1,\dots ,N$.
which converges to 0 as $m\to +\infty $.
which converges exponentially fast to 0 as $m\to +\infty $. □
6.3 Continuity of the membrane potentials
Proposition 5 The deterministic part of the membrane potential, ${\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}(t,\omega ),t,\omega )$, is continuous and its mvariation decays exponentially fast with m.
and ${\Gamma}_{k}({t}_{1},{t}_{2},\omega )$ is continuous in ω.
which converges to 0 as $m\to +\infty $.
and ${var}_{m}[{\mathcal{V}}_{k}^{(\mathrm{syn})}({\tau}_{k}(t,\cdot ),t,\cdot )]$ converges to 0 exponentially fast as $m\to +\infty $.
and ${\mathcal{V}}_{k}^{(\mathrm{ext})}({\tau}_{k}(t,\cdot ),t,\cdot )$ is continuous.
As a conclusion, ${\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}(t,\cdot ),t,\cdot )$ is continuous as the sum of two continuous functions. □
6.4 Continuity of the variance of ${\mathcal{V}}_{k}^{(\mathrm{noise})}({\tau}_{k}(t,\cdot ),t,\cdot )$
Using the same type of arguments, one can also prove that
Proposition 6 The variance${\sigma}_{k}({\tau}_{k}(t,\omega ),t,\omega )$is continuous in ω, for all t, for all$k=1,\dots ,N$.
and continuity follows. □
6.5 Remark
Note that the variation in all quantities considered here is exponentially decaying with a time constant given by $max({\tau}_{kj},{\tau}_{L,k})$. This is physically satisfactory: the loss of memory in the system is controlled by the leak time and the decay of the postsynaptic potential.
7 Statistics of raster plots
7.1 Conditional probability distribution of ${V}_{k}(t)$
Recall that P is the joint distribution of the noise and $\mathbb{E}[]$ 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:
where${\sigma}_{k}^{2}({\tau}_{k}(t,\omega ),t,\omega )$is given by (27).
Moreover, the${V}_{k}(t)$’s, $k=1,\dots ,N$are conditionally independent.
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7.2 The transition probability
We now compute the probability of a spiking pattern at time $n=[t]$, $\omega (n)$, given the past sequence ${\omega}_{\infty}^{n1}$.
Proposition 8 The probability of$\omega (n)$conditionally to${\omega}_{\infty}^{n1}$is given by:
Since the ${V}_{k}(n1)$’s are conditionally Gaussian, with mean ${\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}(n1,\omega ),n1,\omega )$ and variance ${\sigma}_{k}^{2}({\tau}_{k}(n1,\omega ),n1,\omega )$, we directly obtain (47), (48).
Note that since ${\sigma}_{k}({\tau}_{k}(n1,\omega ),n1,\omega )$ is bounded from below by a positive quantity (see (37)), the ratio $\frac{\theta {\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}(n1,\omega ),n1,\omega )}{{\sigma}_{k}({\tau}_{k}(n1,\omega ),n1,\omega )}$ in (48) is defined for all $\omega \in X$. □
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 ${\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}(n1,\omega ),n1,\omega )$ and ${\sigma}_{k}({\tau}_{k}(n1,\omega ),n1,\omega )$. While the integral defining these terms extends from ${\tau}_{k}(n1,\omega )$ to $n1$ where ${\tau}_{k}(n1,\omega )$ can go arbitrary far in the past, the integrand involves the conductance ${g}_{k}(n1,\omega )$ that summarizes an history dating back to $s=\infty $. As a consequence, the probability transitions generate a stochastic process with unbounded memory, thus nonMarkovian. One may argue that this property is a result of our procedure of taking the initial condition in a infinite past $s\to \infty $, to remove the unresolved dependency on ${g}_{k}(s)$ (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 ${g}_{k}(s)$ and the probability distribution of ${V}_{k}(s)$. Or we take $s\to \infty $, removing the initial condition, to the price of considering a nonMarkovian process. Actually, such processes are widely studied in the literature under the name of “chains with complete connections” [43, 46–49] and several important results can be used here. So we adopt the second approach of the alternative. As a byproduct, the knowledge of the Gibbs measure provided by this analysis allows a posteriori to fix the probability distribution of ${V}_{k}(s)$.
For the sake of completeness, we give here the definition of a chain with complete connections (see [43] for more details). For $n\in \mathbb{Z}$, we note ${\mathcal{A}}_{\infty}^{n1}$ the set of sequences ${\omega}_{\infty}^{n1}$ and ${\mathcal{F}}_{\le n1}$ the related σalgebra, while $\mathcal{F}$ is the σalgebra related with $X={\mathcal{A}}^{\mathbb{Z}}$. $\mathcal{P}(X,\mathcal{F})$ is the set of probability measures on $(X,\mathcal{F})$.
such that the following conditions hold for every $n\in \mathbb{Z}$:

For every $\omega (n)\in \mathcal{A}$, the function ${P}_{n}[\omega (n).]$ is measurable with respect to ${\mathcal{F}}_{\le n1}$.

For every ${\omega}_{\infty}^{n1}\in {\mathcal{A}}_{\infty}^{n1}$,$\sum _{\omega (n)\in \mathcal{A}}{P}_{n}[\omega (n){\omega}_{\infty}^{n1}]=1.$
Such a measure μ is called a “chain with complete connections consistent with the system of transition probabilities ${\{{P}_{n}\}}_{n\in \mathbb{Z}}$”.
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 ${P}_{n}[\omega (n){\omega}_{\infty}^{n1}]$ proved below. To simplify notations, we write $p(n,\omega )$ instead of ${P}_{n}[\omega (n){\omega}_{\infty}^{n1}]$ 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 $p(n,\omega )$ with respect to ω follows from the continuity of ${\mathcal{V}}_{k}^{(\mathrm{det})}({\tau}_{k}($