scholarly journals Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

2021 ◽  
Vol Volume 1 ◽  
Author(s):  
Jian-Guo Liu ◽  
Ziheng Wang ◽  
Yantong Xie ◽  
Yuan Zhang ◽  
Zhennan Zhou
Keyword(s):  
Stochastic Process ◽  
Regularization Parameter ◽  
Nonlinear Models ◽  
Mean Field ◽  
Planck Equation ◽  
Iteration Scheme ◽  
Large Network ◽  
Fire Model ◽  
Integrate And Fire ◽  
Whole Space

In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.

scholarly journals Network of interacting neurons with random synaptic weights

2019 ◽  
Vol 65 ◽  
pp. 445-475 ◽  
Author(s):  
Paolo Grazieschi ◽  
Marta Leocata ◽  
Cyrille Mascart ◽  
Julien Chevallier ◽  
François Delarue ◽  
...  
Keyword(s):  
Collective Behavior ◽  
Finite Time ◽  
Mean Field ◽  
Planck Equation ◽  
Fire Model ◽  
Integrate And Fire ◽  
Fokker Planck Equation ◽  
Connection Graph ◽  
Typical Instance ◽  
Theoretical Results

Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.

scholarly journals Event-Driven Simulations of Nonlinear Integrate-and-Fire Neurons

2007 ◽  
Vol 19 (12) ◽  
pp. 3226-3238 ◽  
Author(s):  
Arnaud Tonnelier ◽  
Hana Belmabrouk ◽  
Dominique Martinez
Keyword(s):  
Neural Networks ◽  
Spiking Neural Networks ◽  
Synaptic Currents ◽  
Fire Model ◽  
Integrate And Fire ◽  
Fire Models ◽  
Event Driven

Event-driven strategies have been used to simulate spiking neural networks exactly. Previous work is limited to linear integrate-and-fire neurons. In this note, we extend event-driven schemes to a class of nonlinear integrate-and-fire models. Results are presented for the quadratic integrate-and-fire model with instantaneous or exponential synaptic currents. Extensions to conductance-based currents and exponential integrate-and-fire neurons are discussed.

scholarly journals Generalized Integrate-and-Fire Models of Neuronal Activity Approximate Spike Trains of a Detailed Model to a High Degree of Accuracy

2004 ◽  
Vol 92 (2) ◽  
pp. 959-976 ◽  
Author(s):  
Renaud Jolivet ◽  
Timothy J. Lewis ◽  
Wulfram Gerstner
Keyword(s):  
Cortical Neurons ◽  
Direct Reduction ◽  
Spike Trains ◽  
Model Parameters ◽  
Detailed Model ◽  
Fire Model ◽  
Integrate And Fire ◽  
Fire Models ◽  
Wide Range ◽  
Simple Models

We demonstrate that single-variable integrate-and-fire models can quantitatively capture the dynamics of a physiologically detailed model for fast-spiking cortical neurons. Through a systematic set of approximations, we reduce the conductance-based model to 2 variants of integrate-and-fire models. In the first variant (nonlinear integrate-and-fire model), parameters depend on the instantaneous membrane potential, whereas in the second variant, they depend on the time elapsed since the last spike [Spike Response Model (SRM)]. The direct reduction links features of the simple models to biophysical features of the full conductance-based model. To quantitatively test the predictive power of the SRM and of the nonlinear integrate-and-fire model, we compare spike trains in the simple models to those in the full conductance-based model when the models are subjected to identical randomly fluctuating input. For random current input, the simple models reproduce 70–80 percent of the spikes in the full model (with temporal precision of ±2 ms) over a wide range of firing frequencies. For random conductance injection, up to 73 percent of spikes are coincident. We also present a technique for numerically optimizing parameters in the SRM and the nonlinear integrate-and-fire model based on spike trains in the full conductance-based model. This technique can be used to tune simple models to reproduce spike trains of real neurons.

scholarly journals Stimulus-Dependent Correlations in Threshold-Crossing Spiking Neurons

2009 ◽  
Vol 21 (8) ◽  
pp. 2269-2308 ◽  
Author(s):  
Yoram Burak ◽  
Sam Lewallen ◽  
Haim Sompolinsky
Keyword(s):  
Ganglion Cells ◽  
Retinal Ganglion ◽  
Qualitative Properties ◽  
Fire Model ◽  
Integrate And Fire ◽  
Threshold Crossing ◽  
Cross Correlations ◽  
Bursting Events ◽  
The One ◽  
The Relationship

We consider a threshold-crossing spiking process as a simple model for the activity within a population of neurons. Assuming that these neurons are driven by a common fluctuating input with gaussian statistics, we evaluate the cross-correlation of spike trains in pairs of model neurons with different thresholds. This correlation function tends to be asymmetric in time, indicating a preference for the neuron with the lower threshold to fire before the one with the higher threshold, even if their inputs are identical. The relationship between these results and spike statistics in other models of neural activity is explored. In particular, we compare our model with an integrate-and-fire model in which the membrane voltage resets following each spike. The qualitative properties of spike cross-correlations, emerging from the threshold-crossing model, are similar to those of bursting events in the integrate-and-fire model. This is particularly true for generalized integrate-and-fire models in which spikes tend to occur in bursts, as observed, for example, in retinal ganglion cells driven by a rapidly fluctuating visual stimulus. The threshold-crossing model thus provides a simple, analytically tractable description of event onsets in these neurons.

scholarly journals Adaptive Exponential Integrate-and-Fire Model as an Effective Description of Neuronal Activity

2005 ◽  
Vol 94 (5) ◽  
pp. 3637-3642 ◽  
Author(s):  
Romain Brette ◽  
Wulfram Gerstner
Keyword(s):  
Simple Model ◽  
Expressive Power ◽  
Detailed Model ◽  
Fire Model ◽  
Integrate And Fire ◽  
Conductance States ◽  
Effective Description ◽  
High Conductance

We introduce a two-dimensional integrate-and-fire model that combines an exponential spike mechanism with an adaptation equation, based on recent theoretical findings. We describe a systematic method to estimate its parameters with simple electrophysiological protocols (current-clamp injection of pulses and ramps) and apply it to a detailed conductance-based model of a regular spiking neuron. Our simple model predicts correctly the timing of 96% of the spikes (±2 ms) of the detailed model in response to injection of noisy synaptic conductances. The model is especially reliable in high-conductance states, typical of cortical activity in vivo, in which intrinsic conductances were found to have a reduced role in shaping spike trains. These results are promising because this simple model has enough expressive power to reproduce qualitatively several electrophysiological classes described in vitro.

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