[Paper Review] Noisy threshold in neuronal models: connections with the noisy leaky integrate-and-fire model

Grégory Dumont, Jacques Henry+1 more|arXiv (Cornell University)|Dec 10, 2015

Neural dynamics and brain function21 references5 citations

TL;DR

This paper establishes an explicit integral transform linking the noisy leaky integrate-and-fire (NLIF) model, governed by a Fokker-Planck (FP) equation with absorbing boundary, to the age-structured (AS) escape-rate model with time-dependent threshold noise. The transform maps solutions of the AS system—describing refractory states via time since last spike—into solutions of the FP equation, revealing a mathematical equivalence rooted in survivor functions and Bayes' rule, with the transform's kernel representing the conditional probability of membrane potential given age.

ABSTRACT

Providing an analytical treatment to the stochastic feature of neurons' dynamics is one of the current biggest challenges in mathematical biology. The noisy leaky integrate-and-fire model and its associated Fokker-Planck equation are probably the most popular way to deal with neural variability. Another well-known formalism is the escape-rate model: a model giving the probability that a neuron fires at a certain time knowing the time elapsed since its last action potential. This model leads to a so-called age-structured system, a partial differential equation with non-local boundary condition famous in the field of population dynamics, where the {\it age} of a neuron is the amount of time passed by since its previous spike. In this theoretical paper, we investigate the mathematical connection between the two formalisms. We shall derive an integral transform of the solution to the age-structured model into the solution of the Fokker-Planck equation. This integral transform highlights the link between the two stochastic processes. As far as we know, an explicit mathematical correspondence between the two solutions has not been introduced until now.

Motivation & Objective

• To establish a rigorous mathematical connection between two prominent formalisms for modeling neural stochasticity: the noisy leaky integrate-and-fire (NLIF) model and the age-structured (AS) escape-rate model.
• To resolve the long-standing gap in explicitly linking the Fokker-Planck (FP) solution of the NLIF model with the AS system's solution, which describes refractory states via time since last spike.
• To provide a probability-theoretic interpretation of the transform using Bayes’ rule, showing that the kernel corresponds to P(v|a), the probability of membrane potential v given age a.
• To explore the feasibility of an inverse transform, identifying that time-dependent kernels and compatibility conditions between initial densities are required, with equilibrium assumptions enabling approximate inversion.

Proposed method

• Derive an integral transform that maps the solution of the age-structured (AS) partial differential equation—describing refractory density as a function of age a—into the solution of the Fokker-Planck (FP) equation for the NLIF model.
• Use the survivor function g(t) from renewal theory as a key component of the transform kernel, linking the probability of survival to the threshold crossing time.
• Apply Bayes’ rule to interpret the transform kernel P(v|a) as the conditional probability of membrane potential v given the neuron's age a, with time-invariant dynamics embedded in the age variable.
• Construct a candidate inverse transform under equilibrium assumptions, using steady-state distributions p∞(v) and n∞(a) to define a time-independent version of P(a|v).
• Demonstrate the equivalence numerically via Euler-Maruyama and Gillespie-like simulations, showing identical firing time distributions across both models under identical parameters.
• Establish that the set of solutions generated by the transform forms an attractor set as t → ∞, indicating long-term consistency between the two models.

Experimental results

Research questions

• RQ1 Can an explicit mathematical correspondence be established between the solutions of the Fokker-Planck equation in the noisy leaky integrate-and-fire model and the age-structured system in the escape-rate model?
• RQ2 What is the probabilistic interpretation of the integral transform linking the two models, and how does it relate to conditional probabilities and survivor functions?
• RQ3 Is it possible to construct an inverse transform from the FP solution back to the AS solution, and what constraints or assumptions are required for such an inverse?
• RQ4 How do the initial conditions of the FP and AS systems relate, and what compatibility condition must they satisfy for the transform to be consistent?
• RQ5 What role does the equilibrium distribution play in enabling an approximate inverse transformation, and how does this relate to maximum-entropy assumptions?

Key findings

• The solution to the age-structured (AS) system can be mapped into the solution of the Fokker-Planck (FP) equation via an explicit integral transform whose kernel is the conditional probability P(v|a), representing the likelihood of membrane potential v given age a.
• The transform is grounded in renewal theory and survivor functions, with the kernel being time-independent due to the age variable encoding all necessary temporal information.
• The FP solution is shown to be an attractor of the transformed AS solutions as time t approaches infinity, indicating long-term dynamical consistency between the models.
• An approximate inverse transform is constructed under equilibrium assumptions, using steady-state distributions p∞(v) and n∞(a), enabling a bidirectional mapping in the long-time limit.
• Numerical simulations confirm that both models produce identical firing time distributions when parameters are matched, validating the theoretical equivalence.
• The study reveals that the AS model encodes hidden information about the last spike time through the age variable, which is not explicitly present in the FP model, necessitating compatibility conditions for a full inverse transform.