Stochastic synchronization of neuronal populations with intrinsic and extrinsic noise
© Bressloff, Lai; licensee Springer 2011
Received: 12 November 2010
Accepted: 3 May 2011
Published: 3 May 2011
We extend the theory of noise-induced phase synchronization to the case of a neural master equation describing the stochastic dynamics of an ensemble of uncoupled neuronal population oscillators with intrinsic and extrinsic noise. The master equation formulation of stochastic neurodynamics represents the state of each population by the number of currently active neurons, and the state transitions are chosen so that deterministic Wilson-Cowan rate equations are recovered in the mean-field limit. We apply phase reduction and averaging methods to a corresponding Langevin approximation of the master equation in order to determine how intrinsic noise disrupts synchronization of the population oscillators driven by a common extrinsic noise source. We illustrate our analysis by considering one of the simplest networks known to generate limit cycle oscillations at the population level, namely, a pair of mutually coupled excitatory (E) and inhibitory (I) subpopulations. We show how the combination of intrinsic independent noise and extrinsic common noise can lead to clustering of the population oscillators due to the multiplicative nature of both noise sources under the Langevin approximation. Finally, we show how a similar analysis can be carried out for another simple population model that exhibits limit cycle oscillations in the deterministic limit, namely, a recurrent excitatory network with synaptic depression; inclusion of synaptic depression into the neural master equation now generates a stochastic hybrid system.
Synchronous oscillations are prevalent in many areas of the brain including sensory cortices, thalamus and hippocampus . Recordings of population activity based on the electroencephalogram (EEG) or the local field potential (LFP) often exhibit strong peaks in the power spectrum at certain characteristic frequencies. For example, in the visual system of mammals, cortical oscillations in the γ frequency band (20-70 Hz) are generated with a spatially distributed phase that is modulated by the nature of a visual stimulus. Stimulus-induced phase synchronization of different populations of neurons has been proposed as a potential solution to the binding problem, that is, how various components of a visual image are combined into a single coherently perceived object [2, 3]. An alternative suggestion is that such oscillations provide a mechanism for attentionally gating the flow of neural information [4, 5]. Neuronal oscillations may be generated by intrinsic properties of single cells or may arise through excitatory and inhibitory synaptic interactions within a local population of cells. Irrespective of the identity of the basic oscillating unit, synchronization can occur via mutual interactions between the oscillators or via entrainment to a common periodic stimulus in the absence of coupling.
From a dynamical systems perspective, self-sustained oscillations in biological, physical and chemical systems are often described in terms of limit cycle oscillators where the timing along each limit cycle is specified in terms of a single phase variable. The phase-reduction method can then be used to analyze synchronization of an ensemble of oscillators by approximating the high-dimensional limit cycle dynamics as a closed system of equations for the corresponding phase variables [6, 7]. Although the phase-reduction method has traditionally been applied to deterministic limit cycle oscillators, there is growing interest in extending the method to take into account the effects of noise, in particular, the phenomenon of noise induced phase synchronization [8–15]. This concerns the counterintuitive idea that an ensemble of independent oscillators can be synchronized by a randomly fluctuating input applied globally to all of the oscillators. Evidence for such an effect has been found in experimental studies of oscillations in the olfactory bulb . It is also suggested by the related phenomenon of spike-time reliability, in which the reproducibility of a single neuron’s output spike train across trials is greatly enhanced by a fluctuating input when compared to a constant input [16, 17].
In this paper we extend the theory of noise-induced phase synchronization to the case of a neural master equation describing the stochastic dynamics of an ensemble of uncoupled neuronal population oscillators with intrinsic and extrinsic noise. The master equation formulation of stochastic neurodynamics represents the state of each population by the number of currently active neurons, and the state transitions are chosen such that deterministic Wilson-Cowan rate equations [18, 19] are recovered in an appropriate mean-field limit (where statistical correlations can be neglected) [20–23]. We will consider the particular version of the neural master equation introduced by Bressloff , in which the state transition rates scale with the size N of each population in such a way that the Wilson-Cowan equations are obtained in the thermodynamic limit . Thus, for large but finite N, the network operates in a regime characterized by Gaussian-like fluctuations about attracting solutions (metastable states) of the mean-field equations (at least away from critical points), combined with rare transitions between different metastable states . (In contrast, the master equation of Buice et al. assumes that the network operates in a Poisson-like regime at the population level [21, 22].) The Gaussian-like statistics can be captured by a corresponding neural Langevin equation that is obtained by carrying out a Kramers-Moyal expansion of the master equation . One motivation for the neural master equation is that it represents an intrinsic noise source at the network level arising from finite size effects. That is, a number of studies of fully or sparsely connected integrate-and-fire networks have shown that under certain conditions, even though individual neurons exhibit Poisson-like statistics, the neurons fire asynchronously so that the total population activity evolves according to a mean-field rate equation [26–30]. However, formally speaking, the asynchronous state only exists in the thermodynamic limit , so that fluctuations about the asynchronous state arise for finite N [31–34]. (Finite-size effects in IF networks have also been studied using linear response theory .)
The structure of the paper is as follows. First, we introduce the basic master equation formulation of neuronal population dynamics. We reduce the master equation to a corresponding neural Langevin equation and show that both intrinsic and extrinsic noise sources lead to multiplicative white noise terms in the Langevin equation. We then consider an ensemble of uncoupled neuronal populations each of which evolves according to a neural master equation. We assume that each population supports a stable limit cycle in the deterministic or mean-field limit. We apply stochastic phase reduction and averaging methods to the corresponding system of neural Langevin equations, following along similar lines to Nakao et al. , and use this to determine how independent intrinsic noise disrupts synchronization due to a common extrinsic noise source. (Previous studies have mostly been motivated by single neuronal oscillator models, in which both the independent and common noise sources are extrinsic to the oscillator. In contrast, we consider a stochastic population model in which the independent noise sources are due to finite size effects intrinsic to each oscillator.) We then apply our analysis to one of the simplest networks known to generate limit cycle oscillations at the population level, namely, a pair of mutually coupled excitatory (E) and inhibitory (I) subpopulations . A number of modeling studies of stimulus-induced oscillations and synchrony in primary visual cortex have taken the basic oscillatory unit to be an E-I network operating in a limit cycle regime [37, 38]. The E-I network represents a cortical column, which can synchronize with other cortical columns either via long-range synaptic coupling or via a common external drive. In the case of an E-I network, we show how the combination of intrinsic independent noise and extrinsic common noise can lead to clustering of limit cycle oscillators due to the multiplicative nature of both noise sources under the Langevin approximation. (Clustering would not occur in the case of additive noise.) Finally, we show how a similar analysis can be carried out for another important neuronal population model that exhibits limit cycle oscillations in the deterministic limit, namely, an excitatory recurrent network with synaptic depression; such a network forms the basis of various studies of spontaneous synchronous oscillations in cortex [39–43]. We also highlight how the inclusion of synaptic depression into the master equation formulation leads to a novel example of a stochastic hybrid system .
2 Neural Langevin equation
with gain γ and maximum firing rate . (Any threshold can be absorbed into the external inputs .) Equation (1) preserves the normalization condition for all . The master equation given by equations (1) and (2) is a phenomenological representation of stochastic neurodynamics [20, 23]. It is motivated by various studies of noisy spiking networks which show that under certain conditions, even though individual neurons exhibit Poisson-like statistics, the neurons fire asynchronously so that the population activity can be characterized by fluctuations around a mean rate evolving according to a deterministic mean-field equation [26–29]. On the other hand, if population activity is itself Poisson-like, then it is more appropriate to consider an N-independent version of the master equation, in which and [21, 22]. The advantage of our choice of scaling from an analytical viewpoint is that one can treat as a small parameter and use perturbation methods such as the Kramers-Moyal expansion to derive a corresponding neural Langevin equation .
Strictly speaking, the mean-field description is only valid in the thermodynamic limit , and provided that this limit is taken before the limit . In this paper we are interested in the effects of intrinsic noise fluctuations arising from the fact that each neural subpopulation is finite.
Equation (12) is the neural analog of the well known chemical Langevin equation [46, 47]. (A rigorous analysis of the convergence of solutions of a chemical master equation to solutions of the corresponding Langevin equation in the mean-field limit has been carried out by Kurtz .) It is important to note that the Langevin equation (12) takes the form of an Ito rather than Stratonovich stochastic differential equation (SDE). This distinction will be important in our subsequent analysis.
and is an additional independent Wiener process that is common to all subpopulations. We now have a combination of intrinsic noise terms that are treated in the sense of Ito, and an extrinsic noise term that is treated in the sense of Stratonovich. The latter is based on the physical assumption that external sources of noise have finite correlation times, so that we are considering the external noise to be the zero correlation time limit of a colored noise process.
3 Stochastic synchronization of an ensemble of population oscillators
Langevin equations of the form (18) have been the starting point for a number of recent studies of noise-induced synchronization of uncoupled limit cycle oscillators [9, 11–15]. The one major difference from our own work is that these studies have mostly been motivated by single neuron oscillator models, in which both the independent and common noise sources are extrinsic to the oscillator. In contrast, we consider a stochastic population model in which the independent noise sources are due to finite size effects intrinsic to each oscillator. The reduction of the neural master equation (1) to a corresponding Langevin equation (16) then leads to multiplicative rather than additive noise terms; this is true for both intrinsic and extrinsic noise sources. We will show that this has non-trivial consequences for the noise-induced synchronization of an ensemble of population oscillators. In order to proceed, we carry out a stochastic phase reduction of the full Langevin equations (18), following the approach of Nakao et al.  and Ly and Ermentrout . We will only sketch the analysis here, since further details can be found in these references. We do highlight one subtle difference, however, associated with the fact that the intrinsic noise terms are Ito rather than Stratonovich.
3.1 Stochastic phase reduction
Equation (34) was previously derived by Nakao et al.  (see also ). Here, however, there is an additional contribution to the drift term arising from the fact that the independent noise terms appearing in the full system of Langevin equations (18) are Ito rather than Stratonovich, reflecting the fact that they arise from finite size effects.
3.2 Steady-state distribution for a pair of oscillators
where Γ0 is a normalization constant.
The second general result is that the functions and that determine and according to equations (38) are nontrivial products of the phase resetting curves and terms that depend on the transition rates of the original master equation, see equations (17), (25) and (28). This reflects the fact that both intrinsic and extrinsic noise sources in the full neural Langevin equation (18) are multiplicative rather than additive. As previously highlighted by Nakao et al.  for a Fitzhugh-Nagumo model of a single neuron oscillator, multiplicative noise can lead to additional peaks in the function , which can induce clustering behavior within an ensemble of noise-driven oscillators. In order to determine whether or not a similar phenomenon occurs in neural population models, it is necessary to consider specific examples. We will consider two canonical models of population oscillators, one based on interacting sub-populations of excitatory and inhibitory neurons and the other based on an excitatory network with synaptic depression.
4 Excitatory-inhibitory (E-I) network
4.1 Deterministic network
4.2 Stochastic network and noise-induced synchronization
5 Excitatory network with synaptic depression
So far we have applied the stochastic phase reduction method to a two-population model consisting of mutually interacting excitatory and inhibitory populations. This E-I network is one of the simplest population models known to exhibit limit cycle oscillations in the deterministic limit, and forms the basic module in various studies of stimulus-induced oscillations and synchronization in visual cortex [37, 38]. An even simpler population model known to exhibit limit cycle oscillations is a recurrent excitatory network with synaptic depression. For example, Tabak et al. [39, 40] have analyzed Wilson-Cowan mean-field equations representing a recurrent excitatory network with both slow and fast forms of synaptic depression, and used this to model the dynamics of synchronized population bursts in developing chick spinal cord. These burst oscillations are more robust in the presence of an extrinsic noise source or some form of spatial heterogeneity within the network [50, 51]. An excitatory network with synaptic depression and extrinsic noise has also been used to model transitions between cortical Up and Down states [41–43]. Here we will show how our analysis of noise-induced synchronization of population oscillators based on a Langevin approximation of a neural master equation can be extended to take into account the effects of synaptic depression. In addition to the relevance of synaptic depression in the generation of neural oscillations, it is interesting from a mathematical perspective since the resulting master equation provides a novel example of a so-called stochastic hybrid system [44, 52].
5.1 Deterministic network
5.2 Stochastic network and noise-induced synchronization
The master equation (44) is non-autonomous due to the dependence of the birth rate on , with the latter itself coupled to the associated jump Markov process via the depletion rate . Thus equation (46) is only defined between jumps, during which q evolves deterministically.
The system defined by equations (44)-(46) is an example of a so-called stochastic hybrid model based on a piecewise deterministic process. This type of model has recently been applied to genetic networks  and to excitable neuronal membranes [44, 52, 56]. In the latter case, the hybrid model provides a mathematical formulation of excitable membranes that incorporates the exact Markovian dynamics of single stochastic ion channels. Moreover, the limit theorems of Kurtz  can be adapted to prove convergence of solutions of the hybrid model to solutions of a corresponding Langevin approximation in the limit and finite time, where N is the number of ion channels within the membrane [44, 52].
In this paper we extended the theory of noise-induced synchronization to a stochastic Wilson-Cowan model of neural population dynamics formulated as a neural master equation. We considered two canonical network structures that are known to exhibit limit cycle oscillations in the deterministic limit; an E-I network of mutually interacting excitatory and inhibitory populations, and an excitatory network with synaptic depression. In both cases, we used phase reduction methods and averaging theory to explore the effects of intrinsic noise on the synchronization of uncoupled limit cycle oscillators driven by a common extrinsic noise source. We achieved this by first approximating the neural master equation by a corresponding neural Langevin equation. Such an approximation is reasonable for sufficiently large system size N, and provided that there do not exist other stable attractors of the deterministic system . One important consequence of intrinsic noise is that it broadens the distribution of phase differences. The degree of broadening depends on the term , see equation (39), where N is the system size and depends on the intrinsic dynamics of each uncoupled limit cycle oscillator. Another result our study is that the reduction of the master equation generates multiplicative rather than additive terms in the associated Langevin equation for both intrinsic and extrinsic noise sources. Multiplicative noise can lead to clustering of limit cycle oscillators, as was demonstrated in the case of an ensemble of uncoupled E-I networks.
It is important to point out that the master equation formulation of stochastic neurodynamics developed here and elsewhere [21–24] is a phenomenological representation of stochasticity at the population level. It is not derived from a detailed microscopic model of synaptically coupled spiking neurons, and it is not yet clear under what circumstances such a microscopic model would yield population activity consistent with the master equation approach. Nevertheless, if one views the Wilson-Cowan rate equations [18, 19] as an appropriate description of large-scale neural activity in the deterministic limit, it is reasonable to explore ways of adding noise to such equations from a top-down perspective. One possibility is to consider a Langevin version of the Wilson-Cowan equations involving some form of extrinsic additive white noise [57, 58], whereas another is to view the Wilson-Cowan rate equations as the thermodynamic limit of an underlying master equation that describes the effects of intrinsic noise [20–23]. As we have highlighted in this paper, the latter leads to a multiplicative rather than additive form of noise.
There are a number of possible extensions of this work. First, one could consider more complicated network architectures that generate limit cycle oscillations at the population level. One particularly interesting example is a competitive network consisting of two excitatory populations with synaptic depression (or some other form of slow adaptation) that mutually inhibit each other. Such a network has recently been used to model noise-induced switching during binocular rivalry [59–64]. Binocular rivalry concerns the phenomenon whereby perception switches back and forth between different images presented to either eye [65, 66]. Experimentally, it has been found that the eye dominance time statistics may be fit to a gamma distribution, suggesting that binocular rivalry is driven by a stochastic process . One possibility is that there is an extrinsic source of noise associated with the input stimuli. A number of recent models have examined dominance switching times due to additive noise in a competitive Wilson-Cowan model with additional slow adapting variables [61–63]. On the other hand, Laing and Chow  considered a deterministic spiking neuron model of binocular rivalry in which the statistics of the resulting dominance times appeared noisy due to the aperiodicity of the high-dimensional system’s trajectories. The latter is suggestive of an effective intrinsic noise source within a rate-based population model. A second extension of our work would be to introduce synaptic coupling between the limit cycle oscillators. For example, in the case of E-I networks such coupling could represent intracortical connections between columns in visual cortex [37, 38]. The effects of mutual coupling on noise-induced synchronization has been explored within the context of a pair of coupled conductance-based neurons . Finally, the neural master equation has certain similarities to individual-based models in theoretical ecology, in particular, stochastic urn models of predator-prey systems [68, 69]. Given that predator-prey systems often exhibit limit cycle oscillations and receive extrinsic environmental signals, it would be interesting to extend our results on neuronal population oscillators to explore the effects of demographic noise on the stimulus-induced synchronization of an ensemble of ecological communities.
1One could take the number of neurons in each sub-population to be different provided that they all scaled with N. For example, one could identify the system size parameter N with the mean number of synaptic connections into a neuron in a sparsely coupled network.
Finally, noting that for sufficiently large t (after transients have disappeared), we recover equations (42). In constructing a stochastic version of the network, we will assume that the above mean-field approximation still holds even though the activity variables are now random. See  for a recent discussion of the validity of mean-field approximations in a stochastic network model with synaptic depression.
This publication was based on work supported in part by the National Science Foundation (DMS-0813677) and by Award No KUK-C1-013-4 made by King Abdullah University of Science and Technology (KAUST). PCB was also partially supported by the Royal Society Wolfson Foundation.
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