A mathematical model for short-term plasticity

The RIDC NeuroMat research team has put forward a rigorous mathematical model for short-term plasticity. This type of plasticity has been the object of studies since at least the mid 90s, and recent paper by Antonio Galves, Eva Löcherbach, Christophe Pouzat and Errico Presutti has now proposed a simple probabilistic model describing this phenomenon within a large network of neurons.

Short term synaptic plasticity refers to a change in the strength of the interaction between neurons on timescales which are of the order of milliseconds, that is, comparable to the timescale of the spiking activity of the network. In the model the strength of the interactions changes over time as a function of the residual calcium concentration within the cell.  Residual calcium concentration is a measure of transmitter release, so that the larger the calcium concentration the larger the transmitter release.

The framework that has been advanced by the NeuroMat research team provides a way to describe short-term memory. Informally speaking, for instance as published on Wikipedia, short-term memory (or "primary" or "active memory") is the capacity for holding a small amount of information in mind in an active, readily available state for a short period of time. For example, short-term memory can be used to remember a phone number that has just been recited. In the model short time memory is described as the tendency of the system to keep track of an initial stimulus by staying within a certain region of the space of configurations during a short but macroscopic amount of time before finally being kicked out of this region and relaxing to equilibrium.

A particularly interesting feature of the model is the fact the interaction strength between neurons is now modeled as stochastic
chains evolving in time. This express the fact that the residual calcium concentration changes in time as a random function depending on the spiking activity of each neuron.

Simplification

In the paper, short-term plasticity was considered in a model which is mathematically simpler than the model for systems of interacting chains with memory of variable length introduced by Galves and Löcherbach in 2013.

The major difference is that, in the simpler model, a neuron's action potential is not reseted after spiking. This makes the spiking rate of each neuron a continuous function of the past activity of the pre-synaptic neurons. This continuity simplifies the mathematical analysis of the model. Continuous models have been used by other authors as a rough approximation of the observed spiking activity of a biological system of spiking neurons.

The model introduced in Galves and Löcherbach in 2013 makes the membrane potential of each neuron to be reseted to a basic value after each spike, which introduces a discontinuity of spiking rate. Studying the system with an intrinsic discontinuous feature is one of the challenges the NeuroMat research team will face in the coming period, as stressed by Löcherbach in a recent lecture in São Paulo in which she presented the model for short-term plasticity for the first time.

"What we have so far is a robust step in the right direction. This paper is to the best of our knowledge the first rigorous mathematical proof of short-term plasticity", said Antonio Galves. NeuroMat research efforts to model short-term plasticity are ongoing, including working group sections in Paris and São Paulo.

This piece is part of NeuroMat's Newsletter #61. Read more here

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