On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses

TL;DR

This paper provides a comprehensive mathematical analysis of integrate-and-fire neural networks with conductance-based synapses, revealing the structure of their dynamics, stability, and coding properties under finite spike timing precision.

Contribution

It introduces a novel mathematical framework for analyzing conductance-based integrate-and-fire networks, including the concept of the edge of chaos and a new order parameter for network computation.

Findings

The asymptotic dynamics consist of finitely many stable periodic orbits.

A one-to-one correspondence exists between membrane potentials and spike patterns away from chaos.

The introduced order parameter effectively characterizes the network's computational capacity.

Abstract

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the "edge of chaos", a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely "in the spikes" in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.