User:Aurora166

by Yushi Liu, Cong Liu, Pingbo Lu, David Spade, Ruoxi Xu.

People have long recognized the importance of neuron research, part of which now focuses on proposing a suitable model to characterize and describe the neuron systems. As early as 1907, Lapicque raised the Integrate-and-fire model, which is now viewed as the pioneer work in this field. For this model, the resistor and a capacitor form an electric circuit in parallel. The capacitor is charged until it reaches a certain threshold level, and then it is discharged. This cycle produces the spike of active potential and relaxation of the potential.[1] This allows people to calculate the rate of the spike produced when the neuron is coupled with a fixed-voltage electrode.[1] In 1936, Hill revised the Integrate-and-fire model, allowing the separation of the slow and rapid “spike”.[2] Nowadays, the behind-the-scenes mechanism helps us to understand that the removal of the action potential generating mechanism is powerful to characterize the dynamic behavior of neurons.[3]

For many years, people have focused on improving the Integrate-and-fire model. For example, the synaptic input is stochastic and viewed as a homogenous Possion process.[3] However, when the input is a non-homogenous Possion process, it is very difficult to obtain the analytical solution from the Integrate-and-fire model. The spike response model was motivated by this and was proposed.[4] In this model, the membrane potential is dependent on the time at which the spike occurs.[4] This enables neurons to reduce responsiveness and increase the threshold after resetting the post-spike potential with the addition of one “post-pontential” term.[4] Despite various versions of the modified integrate-and-fire model, they are still limited by the nature of the Integrate-and-fire model as a deterministic model. Random effect(s) need to be introduced into this model. Generally, two random effects are viewed to be inevitable in the characterization of neuron behavior. One random effect is attributed to the stochastic nature of the machinery responsible for the ion channel's opening and its release of the neuron transmitter.[3] The other random effect is attributed to the stochastic synaptic neuron input(i.e. the variation in the time of neuron input arrival).[3] It can be shown that the first source produces comparably insignificant effect when the second variation source is considered.[5]

Here, we briefly discuss some new models that incorporate the traditional Integrate-and-fire model, and the corresponding stochastic processes. They may do a better job of capturing and describing the neuron potential than do previous models.

Diffusion models of neuron

In diffusion models, the membrane potential is considered to be a continuous time and continuous path Markov process (i.e. a diffusion process)[6] Then the synaptic input can be characterized as the following stochastic differential equation:[3]

$\tau {\frac {dv(t)}{dt}}=-[v(t)-V_{0}]+\mu +\sigma {\sqrt {2\tau }}\xi (t)$

Let $<\xi (t)>=0$, $<\xi (t)\xi (t')>=\delta (t-t')$, $v(t)$ represent membrane potential at t, $v(0)$ is the reset potential at time $t=0$. $\mu$ is the mean synaptic input and $\sigma$ is the intensity "correction" for the Brownian motion process $\xi (t)$.[3] In order to solve the above equation, a simpler process is introduced for later use.

Wiener process

The Wiener process was first applied to the leakless Integrate-and-fire neuron model. In this case, we neglect the decay of membrane potential with time.[7] This is the limiting case of a random walk process, during which the membrane potential changes stepwise upon the arrival of the incoming post-synaptic potential.[7] Although the random walk process is discrete, taking the limit forces a smaller stepwise change of membrane potential and a greater rate of synaptic input arrival.[3] Consequently, the membrane potential change becomes continuous(typically known as a diffusion approximation) and can be characterized by the following equation:[3,8]

$v(t)=v_{0}+\mu _{W}t+\sigma _{W}W(t),\quad t>0$

Here, $W(t)$ is a standard Wiener process with drift parameter $\mu _{W}$ and $\sigma _{W}$ as the variance parameter. $v(0)=v_{0}$ and they retain the same meaning as above.[3]

This serves as one of the simplest models. Furthermore, people derive with the combination of Integrate-and-fire model, and the stochastic process. It is a good approximation in the situation of a positive drift and summation of synaptic input over the threshold on a short time scale.[3] However, some of the assumptions and the situation are hard to satisfy. For example, the membrane leakage is unavoidable in almost all situations. This motivated the derivation of the following models.

Ornstein-Uhlenback process and Stein’s model

One stochastic process proposed by Stein is as follows:[3,9]

$\tau _{m}{\frac {dv(t)}{dt}}=[v(t)-V_{0}]+a_{E}S_{E}(t)+a_{I}S_{I}(t)$

This equation is used to characterize the change of membrane potential between two firing events.[3] And $a_{E}>0$ and $a_{I}<0$ represent the change in potential caused by an associated excitatory or inhibitory synaptic input.[3, 9] $S_{E}$ and $S_{I}$ are corresponding Poisson processes with excitatory or inhibitory synaptic inputs.[3] $\tau _{m}$ represents the constant determined by the decay curve between subsequent synaptic inputs.[3] The continuous version of Stein’s model can be derived via the Ornstein-Uhlenbeck process.[3] However, to solve this problem, a diffusion process with the same first and second moments as the Stein’s model needs to be constructed.[10] For a free membrane potential when the spiking threshold is neglected, define the corresponding first and second moments as follows:[3]

$\mu (t,v_{0})\equiv E[v(t)|v_{0},t_{0}=0]$

$\sigma ^{2}(t,v_{0})\equiv Var[v(t)|v_{0},t_{0}=0]$

With the above layout, the solution for the case of current synapses with a homogenous Poisson process input is:[3]

$\mu (t,v_{0})=v_{0}e^{-t/\tau _{m}}+\mu _{Q}(1-e^{-t/\tau _{m}})$

$\sigma ^{2}(t;v_{0})=\sigma _{Q}^{2}(1-e^{-2t/\tau _{m}})$

Compared with Ornstein-Uhlenbeck process, we have:[3]

$\tau =\tau _{m}$

$\mu =\mu _{Q}\equiv \tau _{m}(a_{E}\lambda _{E}-a_{I}\lambda _{I})$

$\sigma ^{2}=\sigma _{Q}^{2}\equiv {\frac {\tau _{m}}{2}}(a_{E}^{2}\lambda _{E}+a_{I}^{2}\lambda _{I})$

For Stein’s model, the diffusion approximation is quite useful for the case of modeling the synaptic input as white noise. Nevertheless, it is limited when the temporal correlation needs to be considered.[3] This will be discussed later.

Gaussian approximation

If we are considering the case of free membrane potential, a Gaussian approximation can be used instead of a diffusion approximation. Here, define a conditional probability, where we have the probabilty of the free membrane potential with value $v$ at time $t$, given that the membrane potential at an earlier time $t_{0}$ was $v_{0}$, as $p(v,t|v_{0},0)$. This conditional probability can be written as:[3,11]

$p(v,t|v_{0},0)={\frac {1}{\sqrt {2\pi \sigma ^{2}(t;v_{0})}}}{\text{exp}}\left\{-{\frac {[v-\mu (t;v_{0})]^{2}}{2\sigma ^{2}(t;v_{0})}}\right\}$

where $\mu (t;v_{0})$ and $\sigma ^{2}(t;v_{0})$ are defined as:

$\mu (t,v_{0})\equiv E[v(t)|v_{0},t_{0}=0]$

$\sigma ^{2}(t,v_{0})\equiv Var[v(t)|v_{0},t_{0}=0]$;

This approximation is extremely useful when many small-amplitude inputs are available. This is a direct result of the central limit theorem.[12] From a simulation result, the Gaussian approximation is useful for the integrate-and-fire model with conductance synapses.[13]

Stein model with conductance

Tuckwell used the following equation to characterize the Stein model with conductance synapse:[3,8,14]

$\tau _{m}{\frac {dv(t)}{dt}}=-[v(t)-V_{0}]+g_{E}[V_{E}-v(t)]S_{E}(t)+g_{I}[V_{I}-v(t)]S_{I}(t)$

Here, $V_{E}$ and $V_{I}$ are reversal potentials. They are generated by the equilibrium potentials of ion channels.[3] In addition, when the membrane potential passes through the reversal potential, the current flow changes its direction. This is different from the original Stein model, in which the discontinuous jump size depends on the particular state of the membrane potential,[3] and the summation of individual post-synaptic potentials(whether excitatory or inhibitory) is nonlinear.[3] $g_{E}$ and $g_{I}$ are two parameters. They mean the following: the (integrated excitatory conductance during the time for the synaptic behavior)/neural capacitance, and (inhibitory conductance over the time for the synaptic behavior)/the neural capacitance, respectively.[3] The use of the Stein model with conductance synapses is limited by the nature of non-linear summation of small synaptic inputs and discontinous membrane potential.

Fokker-Planck formalism

Recall from above that when the noise is correlated, the applicability of the diffusion approximation is limited. Normally in this case, the Fokker-Planck formalism would produce a much better result than a simple diffusion approximation. The basis of this formalism lies in the Fokker-Planck equation. It characterizes the time evolution of the probability density. Here, the probability density is $P(v,t)$, the membrane potential.[3,15]

${\frac {\partial }{\partial t}}P(v,t)=\left[-{\frac {\partial }{\partial v}}A(v)+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial v^{2}}}B(v)\right]P(v,t)$

Here, $A(v)$ represents the drift function and $B(v)$ represents the diffusion function. They are the first two central moments of the distribution for the changes of membrane potential that are caused by the synaptic input.[3] The solution for the drift and diffusion functions in the case of integrate-and-fire neuron model with conductance synapses is:[3]

$A(v)=-{\frac {1}{\tau _{m}}}(v-V_{0})+\lambda _{E}g_{E}(V_{E}-v)+\lambda _{I}g_{I}(V_{I}-v)$

$B(v)=\lambda _{E}g_{E}^{2}(V_{E}-v)^{2}+\lambda _{I}g_{I}^{2}(V_{I}-v)^{2}$

Furthermore, the Fokker-Planck equation provides a stationary distribution for the membrane potential $P_{0}(v)$ as shown by Brunel.[16] This can be viewed as a generalization of the previous simple diffusion process.

In summary, these modifications of the Integrate-and-fire neuron model with the stochastic process offer more flexible ways to characterize the dynamic behavior of the membrane potential. Although some shortcomings still exist (e.g. the model overlooks the spatial effect of the synaptic input and the spiking mechanism), it remains one of the most important and useful models for the exploration of the function of the nerve system.[16]

Reference

1. Lapicque L. 1907. Recherches quantitatives sur l'excitation electrique des nerfs traitee comme une polarizarition. J. Physolp Pathol Gen(Paris) 9:620-35.
2. Hill AV. 1936. Excitation and accommodation in nerve. Proc R Soc B 119:305-355.
3. Burkitt AN. 2006. A review of the intergrate-and-fire neuron model: I. Homogeneous synaptic input.
4. Gersner W. 1995. Time structure of the activity in neural network models. Phys Rev E 51:738-758.
5. Mainen ZF, and Sejnowski TJ. 1995. Reliability of spike timing in neocortical neurons. Science 268:1503-1506.
6. Tuckwell HC. 1988. Introduction to Theoretical Neurobiology. In linear cable theory and dendritic structure, vol. 1. Cambridge University Press, Cambridge.
7. Gerstein GL, and Mandelbrot B. 1964. Random walk models for the spike activity of a signle neuron. Biophys J 4:41-68.
8. Tuckwell HC. 1988. Introduction to Theoretical Neurobiology. In: Nonlinear and stochastic theories, vol. 2. Cambridge University Press, Cambridge.
9. RB, S. 1965. A theoretical analysis of neuronal variability. Biophys J 5:173-194.
10. Ricciardi LM. 1976. Diffusion approximation for a multi-input model neuron. Biol Cybern 24:237-240.
11. Burkitt AN, and Clark GM. 2000. Calculation of interspike intervals for integrate-and-fire neurons with Poisson distribution fo synaptic inputs. Neural Comput 12:1789-1820.
12. Lamperti J. 1966. Probability, 2nd ed. Wiley, New York.
13. Burkitt AN. 2001. Balanced neurons: analysis of leaky integrate-and-fire neurons with reversal potentials. Biol Cybern 85:247-255.
14. Tuckwell HC. 1979. Synaptic transmission in a model for stochastic neural activity. J.Theor. Biol
15. Van Kampen NG. 1992. Stochastic processes in physics and chemistry. North-Holland, Amsterdam.
16. Brunel N. 2000. Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. J.Comput. Neurosci 8:183-208.