A coarse-grained framework for spiking neuronal networks: between homogeneity and synchrony

Abstract

Homogeneously structured networks of neurons driven by noise can exhibit a broad range of dynamic behavior. This dynamic behavior can range from homogeneity to synchrony, and often incorporates brief spurts of collaborative activity which we call multiple-firing-events (MFEs). These multiple-firing-events depend on neither structured architecture nor structured input, and are an emergent property of the system. Although these MFEs likely play a major role in the neuronal avalanches observed in culture and in vivo, the mechanisms underlying these MFEs cannot easily be captured using current population-dynamics models. In this work we introduce a coarse-grained framework which illustrates certain dynamics responsible for the generation of MFEs. By using a new kind of ensemble-average, this coarse-grained framework can not only address the nucleation of MFEs, but can also faithfully capture a broad range of dynamic regimes ranging from homogeneity to synchrony.

Appendices

Appendix A: Spike resolution

As mentioned in Section 3.1, the system of ODEs given by Eq. (1) is not actually well-posed. We need to clarify the dynamics that ensue whenever multiple neurons cross threshold simultaneously (e.g,. as a consequence of an excitatory synaptic event). To ensure that our system is well-posed, we consider our system to be a specific limit of the following conductance-based system:

$$\begin{array}{@{}rcl@{}} \frac{d}{dt}V_{j}^{Q} &=& -\frac{1}{\tau_{V}}\left(V_{j}^Q-V^{L}\right) \\ &&+\sum_k S^{QY} \delta\left(t-T^Y_{j,k}\right) + g_{j}^{Q,E}\left( t\right) +g_{j}^{Q,I}( t), \notag\\ \frac{d}{dt}g_{j}^{Q,E} &=&\sum_{j^{\prime }\mathrm{~of~type~}E}\sum_{k}S^{QE}\alpha _{E}\left( t-T_{j^{\prime },k}^{E}\right), \notag \\ \frac{d}{dt}g_{j}^{Q,I} &=&\sum_{j^{\prime }\mathrm{~of~type~}I}\sum_{k}S^{QI}\alpha _{I}\left( t-T_{j^{\prime },k}^{I}\right), \end{array} $$

(11)

where the $g_{j}^{Q,R}$ model synaptic conductances, and the α _E and α _I are alpha-functions with infinite rise-time and decay times τ _E and τ _I , respectively. For this conductance-based system we also assume that, after any one neuron fires, it is held at V ^L for a time $\tau _{\text {ref}}$, referred to as a refractory period. The well-posed system of ODEs that we use in this work, alluded to by Eq. (1), is actually the system given by Eq. (11) in the limit τ _I , τ _E , $\tau _{\text {ref}} \rightarrow 0$, with $\tau _{I}\ll \tau _{E}\ll \tau _{\text {ref}}$. This is equivalent to the resolution rule used in Refs. (Rangan and Young 2013a) and (Zhang et al. 2013b).The resolution rule we use affects the dynamics within each MFE. For example, if we had instead taken the limit $\tau _{E}\ll \tau _{I}\ll \tau _{\text {ref}} \rightarrow 0$, then the inhibitory neurons would not participate during any MFE, and the dynamics of each MFE could be determined by examining only the excitatory population.

Appendix B: Choosing L ̄

In Section 4.2 we use the PEA-framework to single out MFEs which have magnitude ≥ L ̄ , where L ̄ = 2. MFEs of magnitude less than L ̄ will not be singled out for consideration, and will be lumped into the master-equation evolution of the joint-distribution. In principle, we could have chosen a different number for this ‘MFE-threshold’. When choosing L ̄ , we must balance the following two conflicting goals.

On one hand, if we would like to carefully resolve the dynamics, we should choose L ̄ to be small. Taken to the extreme, setting L ̄ = 1 ensures that we never miss any emergent transients within the network – every firing-event will be resolved and the collection of possible PEA-trajectories will correspond exactly to the ensemble Ω. The drawback to choosing a small L ̄ is that we will frequently be required to resolve synaptic-events, which may be computationally costly.

On the other hand, if we would like to efficiently coarse-grain the dynamics in preparation for dimension-reduction, we should choose L ̄ to be large. If, however, we take L ̄ to be larger than the typical MFE size, then we will miss the majority of the interesting behavior; f _B will reduce to f , and the PEA-framework will reduce to the standard ensemble average.

One way in which we can balance these two goals within our implementation is to choose L ̄ adaptively. Ideally, if the system is currently in a state where l simultaneous excitatory firing-events would often lead to larger MFEs, then we would want L ̄ to be no bigger than l. Given the assumptions we make in Section Section 4.3, we can use this idea to define L ̄ (t) at each time-step by appealing to $\boldsymbol {\rho }_{\textit {single}}(v,t)$. Given $\boldsymbol {\rho }_{\text {single}}$ we can calculate the likelihood $P(L_{E},L_{I}|l ; \boldsymbol {\rho }_{\text {single}}(\cdot ,t))$ that an MFE of magnitude $L_{E},L_{I}$ would be generated, given that l excitatory neurons are driven across threshold by the Poisson input at time t and the other neurons in the network are drawn from $\boldsymbol {\rho }_{\text {single}}(v,t)$ (Zhang et al. 2013b). We then choose L ̄ (t) to be the smallest l for which $P(L_{E},L_{I}|l ; \boldsymbol {\rho }_{\text {single}}(\cdot ,t))$ has a substantial nonzero component for values of L _Q > l. That is to say, we calculate the sums:

$$Z_{l}(t) = \sum\limits_{L_{E} + L_{I} = l+1}^{\infty}P(L_{E},L_{I}|l; \boldsymbol{\rho}_{\text{single}}(\cdot,t)) $$

and choose L ̄ (t) to be the smallest l for which Z _l (t) is non-negligible.Adaptively choosing L ̄ can allow us to efficiently resolve the dynamics of networks which, under drive, switch from one type of regime to another. More details on this algorithm will be reported elsewhere.