Mean-field method for generic conductance-based integrate-and-fire neurons with finite timescales

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Mean-field method for generic conductance-based integrate-and-fire neurons with finite timescales

Marcelo P. Becker and Marco A. P. Idiart

Phys. Rev. E 109, 024406 – Published 14 February 2024

Abstract

The construction of transfer functions in theoretical neuroscience plays an important role in determining the spiking rate behavior of neurons in networks. These functions can be obtained through various fitting methods, but the biological relevance of the parameters is not always clear. However, for stationary inputs, such functions can be obtained without the adjustment of free parameters by using mean-field methods. In this work, we expand current Fokker-Planck approaches to account for the concurrent influence of colored and multiplicative noise terms on generic conductance-based integrate-and-fire neurons. We reduce the resulting stochastic system through the application of the diffusion approximation to a one-dimensional Langevin equation. An effective Fokker-Planck is then constructed using Fox Theory, which is solved numerically using a newly developed double integration procedure to obtain the transfer function and the membrane potential distribution. The solution is capable of reproducing the transfer function and the stationary voltage distribution of simulated neurons across a wide range of parameters. The method can also be easily extended to account for different sources of noise with various multiplicative terms, and it can be used in other types of problems in principle.

Received 19 June 2023
Accepted 8 November 2023

DOI:https://doi.org/10.1103/PhysRevE.109.024406

Physics Subject Headings (PhySH)

Biological neural networks Neuroscience, neural computation & artificial intelligence Noise

Physics of Living SystemsStatistical Physics & Thermodynamics

Authors & Affiliations

Marcelo P. Becker ^*

Bernstein Center for Computational Neuroscience, 10115 Berlin, Germany

Marco A. P. Idiart

Department of Physics, Institute of Physics, Federal University of Rio Grande do Sul, Porto Alegre, Brazil

^*marcelo.becker@bccn-berlin.de

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Issue

Vol. 109, Iss. 2 — February 2024

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Images

Figure 1
Conductance cummulants. Comparison of the analytical expressions obtained for the statistics of $g_{E}$ using the diffusion approximation (lines) with simulations (symbols). The analytical expressions are good descriptions of the simulations for all the range of parameters tested for the first and second moments. The skewness, however, exhibits a deviation from the Gaussian approximation for small $τ_{E}$ . Parameters for first column, $w_{E} = 0.1, w_{I} = 0.4$ ; second column, $w_{I} = 0.8, ν_{i} = 5 H z$
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Figure 2
CoBaIF model with effective time-constant approximation. Comparison of the analytical model (lines) with simulations (symbols) for the simple conductance-based integrate-and-fire neuron using the effective time-constant approximation. In columns (a) and (b), we use three different values of input firing rate $ν_{i}$ as a function of the excitatory time constant and set $w_{E} = 0.1$ and $w_{I} = 0.4$ . In columns (c) and (d), we compare three values of inhibitory weights $w_{I}$ and set $w_{E} = 0.5, ν_{i} = 5 Hz$ . In the first row and columns (a) and (c) the mean potential for a thresholdless model is plotted. Columns (b) and (d) show the standard deviation of the membrane potential for the same thresholdless model. In the second row, columns (a) and (c) display the firing rates for the model with the continuous distribution assumption, and columns (b) and (d) display the firing rates using the double integration procedure. The absolute errors for the corresponding models are plotted in the third row.
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Figure 3
CoBaIF model with full multiplicative treatment. We compare the analytical model with simulations for the simple conductance-based integrate-and-fire neuron, incorporating full multiplicative noise. In columns (a) and (b), we vary the input firing rate $ν_{i}$ as a function of the excitatory time constant and set $w_{E} = 0.1$ and $w_{I} = 0.4$ . In columns (c) and (d), we compare different values of inhibitory weight $w_{I}$ and set $w_{E} = 0.5, ν = 5 H z$ . In the first row, columns (a) and (c) display the firing rate for the model assuming continuous distribution, and columns (b) and (d) display the firing rates using the double integration procedure. The absolute errors for the corresponding models are plotted on the second row.
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Figure 4
Membrane potential distributions for the CoBaIF model. The stationary probability distributions for the Voltage of conductance-based integrate-and-fire neurons. Column (a) $τ_{E} = 1 ms$ , (b) $τ_{E} = 5 ms$ , (c) $τ_{E} = 10 ms$ , (d) $τ_{E} = 20 ms$ , (e) $τ_{E} = 70 ms$ . The first row corresponds to blue curves (full line) in Fig. 2, that is, $ν_{i} = 5 Hz$ and parameters from Table 1. The second row corresponds to the yellow curves (dotted line) on the same figure ( $w_{I} = 10, w_{E} = 5$ , and $ν_{i} = 5 Hz$ ). The different lines correspond to different assumptions in the model, as can be seen in the legend in column (e).
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Figure 5
NMDA model. Comparison of the NMDA analytical model with simulations for three different values of input rate $ν_{I}$ [columns (a) and (b)] and inhibitory synaptic weight $w_{I}$ [columns (c) and (d)] as a function of the interpolation parameter $α$ . In (a) and (b), we set $w_{E} = 0.1$ and $w_{I} = 0.4$ , while in (c) and (d), we set $w_{E} = 0.5$ and $ν_{i} = 5 Hz$ . Columns (a) and (c) assume the continuity of the distribution, while (b) and (d) use the double integration procedure. We calculate the error as the absolute distance between the simulation and analytical results.
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Figure 6
Membrane potential distribution for the NMDA model. We present the stationary probability distribution for different values of the interpolation parameter $α$ : (a) $α = 0$ , (b) $α = 0.1$ , (c) $α = 0.2$ , (d) $α = 0.5$ , and (e) $α = 0.9$ . The first row corresponds to the blue curves (full line) in Fig. 5, with $ν_{i} = 5 Hz$ and parameters from Table 2. The second row corresponds to the yellow curve (dotted line) on the same figure, with $w_{I} = 10, w_{E} = 5$ , and $ν_{i} = 5 Hz$ . The legend indicates different assumptions in the model for each line.
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