Estimating neural connection strengths from firing intervals

TL;DR

This paper presents a mathematically transparent method to estimate neural connection strengths from firing interval data using a nonlinear model, simplifying the inverse problem to solving linear algebraic equations.

Contribution

It introduces a novel approach that reduces the complex inverse problem to solving decoupled ODEs and algebraic equations, with proven continuity under certain conditions.

Findings

The inverse problem can be solved via standard mathematical tools.

The forward operator is shown to be continuous under specific assumptions.

Numerical experiments validate the theoretical approach.

Abstract

We propose and analyse a procedure for using a standard activity-based neuron network model and firing data to compute the effective connection strengths between neurons in a network. We assume a Heaviside response function, that the external inputs are given and that the initial state of the neural activity is known. The associated forward operator for this problem, which maps given connection strengths to the time intervals of firing, is highly nonlinear. Nevertheless, it turns out that the inverse problem of determining the connection strengths can be solved in a rather transparent manner, only employing standard mathematical tools. In fact, it is sufficient to solve a system of decoupled ODEs, which yields a linear system of algebraic equations for determining the connection strengths. The nature of the inverse problem is investigated by studying some mathematical properties of the aforementioned linear system and by a series of numerical experiments. Finally, under an assumption preventing the effective contribution of the network to each neuron from staying at zero, we prove that the involved forward operator is continuous. Sufficient criteria on the external input ensuring that the needed assumption holds are also provided.