View source for Draft:Conductance-Based Refractory Density model

{{Short description|Mathematical model of neuronal population}}
{{Draft topics|biology}}
{{AfC topic|other}}
{{AfC submission|||ts=20241216104855|u=Antonchizhov4wiki|ns=118}}
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The '''Conductance-Based Refractory Density (CBRD) approach''' enables the derivation of a macroscopic model of a neuronal ensemble based on the microscopic dynamics of individual neurons, specifically those described by the [[Hodgkin–Huxley model|Hodgkin-Huxley]]-type framework.<ref>{{Cite journal |last1=Schwalger |first1=Tilo |last2=Chizhov |first2=Anton V |date=2019-10-01 |title=Mind the last spike — firing rate models for mesoscopic populations of spiking neurons |url=https://linkinghub.elsevier.com/retrieve/pii/S095943881930039X |journal=Current Opinion in Neurobiology |series=Computational Neuroscience |volume=58 |pages=155–166 |doi=10.1016/j.conb.2019.08.003 |pmid=31590003 |arxiv=1909.10007 |issn=0959-4388}}</ref> Here, a neuronal ensemble (or neuronal population) refers to as a large (ideally infinite) set of uncoupled stochastic neurons. These neurons receive common input in the form of synaptic current or synaptic conductances. The output signal is the population firing rate.

== Single neuron ==
As a microscopic model of a single neuron, we consider the equations for the neuronal membrane potential  <math>U</math> and gating variables <math>x, y</math> in the following form:

<math>C ~{ dU \over  dt} =-I_L-I_{voltage-gated}-I_{synaptic}+\sigma ~\xi(t)</math>

<math>{dx \over dt}={x_{\infty}(U)-x \over \tau_x(U)}</math>

<math>{dy \over dt}={y_{\infty}(U)-y \over \tau_y(U)}</math>

where <math>I_L, ~I_{voltage-gated}</math> and <math>I_{synaptic}</math> are the leak, voltage-gated and synaptic currents, respectively; <math>\sigma~\xi(t)</math> represents Gaussian white noise with amplitude <math>\sigma</math>; <math>C</math> is the membrane capacitance; <math>x_{\infty}, ~y_{\infty}, ~\tau_x ~\text{and} ~\tau_y</math> are the Hodgkin-Huxley-like approximations of voltage-dependent characteristics of activation and inactivation characteristics for all channels considered.

== Population ==
In the CBRD approach, these equations are parameterized with '''the time elapsed since the last spike''' <math>t^{*}</math> (referred to as age). According to this model, neurons move in the <math>t^{*}</math>-space (Figure) at a unity speed. If the membrane potential of a neuron approaches the threshold <math>V^T</math>, then with a certain probability <math>H</math>, the neuron generates a spike and resets to <math>t^{*}=0</math>, representing the state immediately after the spike.
[[File:CBRD scheme.png|thumb|'''CBRD''' model describes neuronal evolution in the space of '''the time elapsed since the last spike'''.]]
The CBRD model describes the evolution of neurons in the space of the time elapsed since the last spike. The ordinary differential equations are reformulated into transport equations. The transport equation for the neuronal density, referred to as '''the refractory density'''<ref>{{Cite book |title=Spiking neuron models: single neurons, populations, plasticity |date=2002 |publisher=Cambridge University Press |isbn=978-0-511-07817-0 |editor-last=Gerstner |editor-first=Wulfram |location=Cambridge, U.K New York |editor-last2=Kistler |editor-first2=Werner M.}}</ref> <math>\rho(t,t^{*})</math>, completes the system of equations. Here, the probability of firing, represented by '''the hazard-function''' <math>H(U(t,t^{*}), \dot U(t,t^{*}))</math>, depends on the neuronal membrane potential and its derivative in time. The final equations of the CBRD model are one-dimensional transport equations, with the right-hand sides remaining the same as in the original equations:

* for the neuronal (refractory) density:

<math>{\partial \rho \over \partial t} + {\partial \rho \over \partial t^{*}}=-\rho H</math>

* for the membrane potential:

<math>C \left( {\partial U \over \partial t} + {\partial U \over \partial t^{*}} \right)=-I_L-I_{voltage-gated}-I_{synaptic}</math>

* for the gating variables:

<math>{\partial x \over \partial t} + {\partial x \over \partial t^{*}}={x_{\infty}(U)-x \over \tau_x(U)}</math>

<math>{\partial y \over \partial t} + {\partial y \over \partial t^{*}}={y_{\infty}(U)-y \over \tau_y(U)}</math>

The firing rate <math>\nu(t)</math> is obtained from the conservation of the total number of neurons:

<math>\nu(t) \equiv \rho(t,t^{*})\vert_{t^{*}=0}=\int_{+0}^{\infty}\rho H~dt^{*} </math>

The approximation of the hazard function <math>H(U(t,t^{*}), \dot U(t,t^{*}))</math> for the case of white Gaussian noise is derived by solving the Fokker-Planck equation for voltage fluctuations near an arbitrary time-depending mean voltage  <math>U(t,t^{*})</math><ref>{{Cite journal |last=Chizhov and Graham |date=2007 |title=Population model of hippocampal pyramidal neurons, linking a refractory density approach to conductance-based neurons |url=https://journals.aps.org/pre/abstract/10.1103/PhysRevE.75.011924 |journal=Physical Review E |volume=75 |issue=1 |page= 011924|doi=10.1103/PhysRevE.75.011924|pmid= 17358201|bibcode=2007PhRvE..75a1924C }}</ref>:

<math>H(U, \dot U)=A+B,

</math>  where  <math>A= {g_L \over C} ~e^{(0.0061-1.12 T - 0.257 T^2 - 0.072 T^3 - 0.0117 T^4)},
</math>  <math>B= {2 \over {\sqrt{\pi} ~\sigma }} \left[ {\dot U } \right]_{+} {\exp(-T^2) \over {1+\hbox{erf}(T)}},
</math>   <math>T={V^T-U \over {\sigma}} </math>,

where   <math>V^T</math> is the threshold and <math>g_L</math> is the input (leak) conductance;   <math>[~]_{+}</math> is the Heaviside step function.
[[File:CBRD for Wiki.png|thumb|396x396px|Simulation with CBRD model. Top to bottom: input, membrane potential of a representative neuron, firing rate calculated with the Monte-Carlo method (N=1000) and the CBRD model]]
In simulations, the firing rate reflects the synchronization of neurons in transient states  (Figure). The solutions for the firing rate calculated using the CBRD model converge to the results of Monte-Carlo simulations of single neurons.

A realization of the CBRD model, based on a specific realistic conductance-based neuron model<ref>{{Cite journal |last1=Chizhov |first1=Anton V. |last2=Amakhin |first2=Dmitry V. |last3=Sagtekin |first3=A. Erdem |last4=Desroches |first4=Mathieu |date=2023-12-01 |title=Single-compartment model of a pyramidal neuron, fitted to recordings with current and conductance injection |url=https://link.springer.com/article/10.1007/s00422-023-00976-7 |journal=Biological Cybernetics |language=en |volume=117 |issue=6 |pages=433–451 |doi=10.1007/s00422-023-00976-7 |pmid=37755465 |issn=1432-0770}}</ref> is available as Python code.<ref>{{Cite journal |title=CBRD model based on a neuron |url=https://gitlab.inria.fr/anton.chizhov/cbrd_with_my_simple_neuron/-/tree/main |access-date=2024-12-13 |journal=Biological Cybernetics |date=2023 |doi=10.1007/s00422-023-00976-7 |pmid=37755465 |language=en |last1=Chizhov |first1=A. V. |last2=Amakhin |first2=D. V. |last3=Sagtekin |first3=A. E. |last4=Desroches |first4=M. |volume=117 |issue=6 |pages=433–451 }}</ref>

== Generalizations ==
The approach has been generalized for the following cases:

* Colored noise
* Lognormal distribution of weights for the input current
* Finite-size population<ref>{{Cite journal |last1=Schwalger |first1=Tilo |last2=Deger |first2=Moritz |last3=Gerstner |first3=Wulfram |date=2017-04-19 |title=Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size |journal=PLOS Computational Biology |language=en |volume=13 |issue=4 |pages=e1005507 |doi=10.1371/journal.pcbi.1005507 |doi-access=free |issn=1553-7358 |pmc=5415267 |pmid=28422957|arxiv=1611.00294 |bibcode=2017PLSCB..13E5507S }}</ref><ref>{{Cite journal |last1=Dumont |first1=Grégory |last2=Payeur |first2=Alexandre |last3=Longtin |first3=André |date=2017-08-07 |title=A stochastic-field description of finite-size spiking neural networks |journal=PLOS Computational Biology |language=en |volume=13 |issue=8 |pages=e1005691 |doi=10.1371/journal.pcbi.1005691 |doi-access=free |issn=1553-7358 |pmc=5560761 |pmid=28787447|bibcode=2017PLSCB..13E5691D }}</ref>
* Two-compartment neurons
* Bursting neurons
* Alternative approximations of the hazard-function<ref>{{Cite journal |last=Schwalger |first=Tilo |date=2021-10-01 |title=Mapping input noise to escape noise in integrate-and-fire neurons: a level-crossing approach |journal=Biological Cybernetics |language=en |volume=115 |issue=5 |pages=539–562 |doi=10.1007/s00422-021-00899-1 |issn=1432-0770 |pmc=8551127 |pmid=34668051}}</ref>

== Applications ==
The CBRD model has been used to simulate cortical neural tissue activity, where neuronal populations are coupled population-to-population.

== References ==
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