- 5.5.1 Stochastic spike arrival
- 5.5.1.1 Example: Membrane potential fluctuations
- 5.5.1.2 Example: Balanced excitation and inhibition

- 5.5.2 Diffusion limit (*)

- 5.5.3 Interval distribution

5.5 Diffusive noise

The integrate-and-fire model is, in its simplest form, defined by a
differential equation
d*u*/d*t* = - *u* + *R* *I*(*t*) where
is the membrane time constant, *R* the input resistance, and *I* the input
current. The standard procedure of implementing noise in such a differential
equation is to add a `noise term', (*t*), on the right-hand side. The noise
term is a stochastic process called `Gaussian white noise' characterized
by its expectation value,
(*t*) = 0, and the autocorrelation

where is the amplitude of the noise and the membrane time constant of the neuron. The result is a

i.e., an equation for a stochastic process (Ornstein-Uhlenbeck process); cf. van Kampen (1992).

The neuron is said to fire an action potential whenever the membrane
potential *u* reaches the threshold ; cf.
Fig. 5.12. We will refer to Eq. (5.73) as the Langevin
equation of the noisy integrate-and-fire model. The analysis of
Eq. (5.73) in the presence of the threshold is the topic of
this section. Before we start with the discussion of Eq. (5.73), we
indicate how the noise term (*t*) can be motivated by stochastic spike
arrival.

A typical neuron, e.g., a pyramidal cell in the vertebrate cortex, receives input spikes from thousands of other neurons, which in turn receive input from their presynaptic neurons and so forth; see Fig. 5.13. It is obviously impossible to incorporate all neurons in the brain into one huge network model. Instead, it is reasonable to focus on a specific subset of neurons, e.g., a column in the visual cortex, and describe input from other parts of the brain as a stochastic background activity.

Let us consider an integrate-and-fire neuron that is part of a large network.
Its input consists of (i) an external input
*I*^{ext}(*t*); (ii) input
spikes *t*_{j}^{(f)} from other neurons *j* of the network; and (iii) stochastic
spike arrival *t*_{k}^{(f)} due to the background activity in other parts of the
brain. The membrane potential *u* evolves according to

where is the Dirac function and

In Stein's model, each input spike generates a postsynaptic potential
*u*(*t*) = *w*_{j}(*t* - *t*_{j}^{(f)}) with
(*s*) = *e*^{-s/} (*s*),
i.e., the potential jumps upon spike arrival by an amount *w*_{j} and decays
exponentially thereafter. It is straightforward to generalize the model so as
to include a synaptic time constant and work with arbitrary postsynaptic
potentials
(*s*) that are generated by stochastic spike arrival; cf. Fig. 5.14A.

We consider stochastic spike arrival at rate . Each input spike evokes a
postsynaptic potential
*w*_{0} (*s*). The input statistics is assumed
to be Poisson, i.e., firing times are independent. Thus, the input spike
train,

that arrives at neuron

and autocorrelation

cf. Eq. (5.36).

We suppose that the input is weak so that the neuron does not reach its firing
threshold. Hence, we can savely neglect both threshold and reset. Using the
definition of the random process *S* we find for the
membrane potential

We are interested in the mean potential
*u*_{0} = *u*(*t*) and the
variance
*u*^{2} = [*u*(*t*) - *u*_{0}]^{2}. Using
Eqs. (5.76) and (5.77) we find

and

In Fig. 5.14 we have simulated a neuron which receives input from

Mean and fluctuations for Stein's model can be derived by evaluation of
Eqs. (5.79) and (5.80) with
(*s*) = *e*^{-s/}. The result is

u_{0} |
= | w_{0} |
(5.81) |

u^{2} |
= | 0.5 w_{0}^{2} |
(5.82) |

Note that with excitation alone, as considered here, mean and variance cannot be changed independently. As we will see in the next example, a combination of excitation and inhibition allows us to increase the variance while keeping the mean of the potential fixed.

5.5.1.2 Example: Balanced excitation and inhibition

Let us suppose that an integrate-and-fire neuron defined by
Eq. (5.74) with = 10ms receives input from 100 excitatory
neurons (
*w*_{k} = + 0.1) and 100 inhibitory neurons (
*w*_{k} = - 0.1). Each
background neuron *k* fires at a rate of = 10Hz. Thus, in each
millisecond, the neuron receives on average one excitatory and one inhibitory
input spike. Each spike leads to a jump of the membrane potential of ±0.1. The trajectory of the membrane potential is therefore similar to that
of a random walk; cf. Fig. 5.15A. If, in
addition, a constant stimulus
*I*^{ext} = *I*_{0} > 0 is applied so that the
mean membrane potential (in the absence of the background spikes) is just
below threshold, then the presence of random background spikes may drive *u*
towards the firing threshold. Whenever
*u*, the membrane
potential is reset to *u*_{r} = 0.

We note that the mean of the stochastic background input vanishes since
*w*_{k} = 0. Using the same arguments as in the previous
example, we can convince ourselves that the stochastic arrival of background
spikes generates fluctuations of the voltage with variance

u^{2} = 0.5 w_{k}^{2} = 0.1 ; |
(5.83) |

cf. Section 5.5.2 for a different derivation. Let us now increase all rates by a factor of

Since firing is driven by the fluctuations of the membrane potential, the interspike intervals vary considerably; cf. Fig. 5.15. Balanced excitatory and inhibitory spike input, could thus contribute to the large variability of interspike intervals in cortical neurons (Shadlen and Newsome, 1998; van Vreeswijk and Sompolinsky, 1996; Brunel and Hakim, 1999; Amit and Brunel, 1997a; Shadlen and Newsome, 1994; Tsodyks and Sejnowski, 1995); see Section 5.6.

5.5.2 Diffusion limit (*)

In this section we analyze the model of stochastic spike arrival defined in
Eq. (5.74) and show how to map it to the diffusion
model defined in Eq. (5.73)
(Johannesma, 1968; Gluss, 1967; Capocelli and Ricciardi, 1971).
Suppose that the neuron has fired its last spike
at time . Immediately after the firing the membrane potential was
reset to *u*_{r}. Because of the stochastic spike arrival, we cannot predict the
membrane potential for *t* > , but we can calculate its probability
density, *p*(*u*, *t*).

For the sake of simplicity, we set for the time being
*I*^{ext} = 0 in Eq. (5.74). The input spikes at synapse *k*
are generated by a Poisson process and arrive stochastically with rate
(*t*). The probability that no spike arrives in a short time interval
*t* is therefore

If no spike arrives in [

We will refer to

We put Eq. (5.85) in (5.86). To perform the integration, we have to recall the rules for functions, viz., (

Since

For

where we have neglected terms of order

(t) = (t) w_{k}^{2} . |
(5.90) |

The specific process generated by the Langevin-equation (5.73) with

For the transition from Eq. (5.88) to (5.89) we have suppressed higher-order terms in the expansion. The missing terms are

with

The Fokker-Planck equation (5.89) and the Langevin equation (5.73) are equivalent descriptions of drift and diffusion of the membrane potential. Neither of these describe spike firing. To turn the Langevin equation (5.73) into a sensible neuron model, we have to incorporate a threshold condition. In the Fokker-Planck equation (5.89), the firing threshold is incorporated as a boundary condition

Before we continue the discussion of the diffusion model in the presence of a threshold, let us study the solution of Eq. (5.89) without threshold.

In the absence of a threshold (
), both the Langevin
equation (5.73) and the Fokker-Planck equation (5.89) can
be solved. Let us consider Eq. (5.73) for constant . At
*t* = the membrane potential starts at a value *u* = *u*_{r} = 0. Since
(5.73) is a linear equation, its solution is

Since (

In particular, for constant input current

with

The fluctuations of the membrane potential have variance
*u*^{2} = [*u*(*t*|) - *u*_{0}(*t*)]^{2} with *u*_{0}(*t*) given by
Eq. (5.94). The variance can be evaluated with the help of
Eq. (5.93), i.e.,

We use (

Hence, noise causes the actual membrane trajectory to drift away from the noiseless reference trajectory

The solution of the Fokker-Planck equation (5.89) with initial
condition
*p*(*u*,) = (*u* - *u*_{r}) is a Gaussian with mean *u*_{0}(*t*) and
variance
*u*^{2}(*t*), i.e.,

as can be verified by inserting Eq. (5.99) into (5.89); see Fig. 5.16. In particular, the stationary distribution that is approached in the limit of

which describes a Gaussian distribution with mean

Let us consider a neuron that starts at time with a membrane potential
*u*_{r}
and is driven for *t* > by a known input *I*(*t*).
Because of the diffusive noise generated by stochastic spike arrival,
we cannot predict the exact value of the neuronal membrane potential *u*(*t*)
at a later time *t* > ,
but only the probability
that the membrane potential is in a certain interval [*u*_{0}, *u*_{1}].
Specifically, we have

Probu_{0} < u(t) < u_{1} | u() = u_{r} = p(u, t) du |
(5.101) |

where

is the probability that the membrane potential has not reached the threshold. In view of Eq. (5.5), the input-dependent interval distribution is therefore

We recall that

For constant input *I*_{0} the mean interspike interval is
*s* = *s* *P*_{I0}(*s*| 0)d*s* = *s* *P*_{0}(*s*)d*s*; cf.
Eq. (5.21). For the diffusion model Eq. (5.73) with threshold
reset potential *u*_{r}, and membrane time constant , the
mean interval is

where

We have seen that, in the absence of a threshold, the Fokker-Planck Equation
(5.89) can be solved; cf. Eq. (5.99). The transition
probability from an arbitrary starting value *u'* at time *t'* to a new value
*u* at time *t* is

with

A method due to Schrödinger uses the solution of the unbounded problem in order to calculate the input-dependent interval distribution

This integral equation can be solved numerically for the distribution

Cambridge University Press, 2002

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