# Firing rate distributions in plastic networks of spiking neurons

## TL;DR

This paper extends mean-field theory to model how plasticity affects firing rates and connections in complex neuronal networks.

## Contribution

The work introduces a mean-field model that combines structural heterogeneity and synaptic plasticity in spiking neuron networks.

## Key findings

- The model provides exact solutions for firing rate and synaptic weight distributions in plastic networks.
- Simulations confirm the model's accuracy across a practical range of degradation rates.
- The approach reveals how plasticity modulates both activity and structure in neuronal networks.

## Abstract

In recurrent networks of leaky integrate-and-fire neurons, the mean-field theory has been instrumental in capturing the statistical properties of neuronal activity, like firing rate distributions. This theory has been applied to networks with either homogeneous synaptic weights and heterogeneous connections per neuron or vice versa. Our work expands mean-field models to include networks with both types of structural heterogeneity simultaneously, particularly focusing on those with synapses that undergo plastic changes. The model introduces a spike trace for each neuron, a variable that rises with neuron spikes and decays without activity, influenced by a degradation rate rp and the neuron’s firing rate ν. When the ratio α = ν/rp is significantly high, this trace effectively estimates the neuron’s firing rate, allowing synaptic weights at equilibrium to be determined by the firing rates of connected neurons. This relationship is incorporated into our mean-field formalism, providing exact solutions for firing rate and synaptic weight distributions at equilibrium in the high α regime. However, the model remains accurate within a practical range of degradation rates, as demonstrated through simulations with networks of excitatory and inhibitory neurons. This approach sheds light on how plasticity modulates both activity and structure within neuronal networks, offering insights into their complex behavior.

Networks of spiking neurons are complex systems where the structure of connections and the activity patterns generated are deeply intertwined, a relationship often studied using mathematical approaches like the mean-field theory. However, previous studies have primarily focused on networks with limited structural variability, where either the connection strength is nearly identical across the network or the number of connections varies little from one neuron to another. This work takes a step forward by combining both types of structural variability and allowing connection strengths to adapt over time, thereby providing an extended mean-field theory. We derive exact solutions for the distribution of spiking rates and connection strengths at equilibrium and demonstrate their accuracy through numerical simulations, even beyond the defining parameter ranges, offering a more comprehensive and realistic perspective on the interplay between activity and structure in neuronal networks.