Dimensional reduction in networks of non-Markovian spiking neurons: Equivalence of synaptic filtering and heterogeneous propagation delays.

Bard Ermentrout,Maurizio Mattia,Matteo Biggio,Marco Storace,Andrea Galluzzi

doi:10.1371/journal.pcbi.1007404

Bard Ermentrout, Maurizio Mattia + Show 3 more

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https://doi.org/10.1371/journal.pcbi.1007404

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Abstract

Message passing between components of a distributed physical system is non-instantaneous and contributes to determine the time scales of the emerging collective dynamics. In biological neuron networks this is due in part to local synaptic filtering of exchanged spikes, and in part to the distribution of the axonal transmission delays. How differently these two kinds of communication protocols affect the network dynamics is still an open issue due to the difficulties in dealing with the non-Markovian nature of synaptic transmission. Here, we develop a mean-field dimensional reduction yielding to an effective Markovian dynamics of the population density of the neuronal membrane potential, valid under the hypothesis of small fluctuations of the synaptic current. Within this limit, the resulting theory allows us to prove the formal equivalence between the two transmission mechanisms, holding for any synaptic time scale, integrate-and-fire neuron model, spike emission regimes and for different network states even when the neuron number is finite. The equivalence holds even for larger fluctuations of the synaptic input, if white noise currents are incorporated to model other possible biological features such as ionic channel stochasticity.

Highlights

Large distributed systems like brain neuronal networks often have to satisfy both timing and space constraints, irrespective of their size
Understanding the collective behavior of the intricate web of neurons composing a brain is one of the most challenging and complex tasks of modern neuroscience. Part of this complexity resides in the distributed nature of the interactions between the network components: for instance, the neurons transmit their messages with delays, which are due to different axonal lengths and/or noninstantaneous synaptic transmission
Starting from the population density dynamics of single-neuron state variables under mean-field approximation, we extend to the colored-noise case the spectral expansion of the associated Fokker-Planck (FP) equation previously derived under whitenoise assumption [15]

Summary

Introduction

Large distributed systems like brain neuronal networks often have to satisfy both timing and space constraints, irrespective of their size. Even in the simplest modeling condition of pointlike spiking neurons, the microscopic dynamics of their state variable (i.e., the membrane potential V(t)) is non-Markovian, and multi-dimensional through Markovian embedding [2, 3]. This microscopic dynamics spans a wide variety of time scales, affecting the stability of various network dynamics [4], the selectivity in transmitting information [5] and the reactivity to suddenly appearing exogenous stimuli [6]

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