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5.5 Diffusive noise

The integrate-and-fire model is, in its simplest form, defined by a differential equation $ \tau_{m}^{}$du/dt = - u + R I(t) where $ \tau_{m}^{}$ 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', $ \xi$(t), on the right-hand side. The noise term $ \xi$ is a stochastic process called `Gaussian white noise' characterized by its expectation value, $ \langle$$ \xi$(t)$ \rangle$ = 0, and the autocorrelation

$\displaystyle \langle$$\displaystyle \xi$(t$\displaystyle \xi$(t')$\displaystyle \rangle$ = $\displaystyle \sigma^{2}_{}$ $\displaystyle \tau_{m}^{}$ $\displaystyle \delta$(t - t') , (5.72)

where $ \sigma$ is the amplitude of the noise and $ \tau_{m}^{}$ the membrane time constant of the neuron. The result is a stochastic differential equation,

$\displaystyle \tau_{m}^{}$$\displaystyle {\frac{{{\text{d}}}}{{{\text{d}}t}}}$u(t) = - u(t) + R I(t) + $\displaystyle \xi$(t) , (5.73)

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 $ \vartheta$; 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 $ \vartheta$ is the topic of this section. Before we start with the discussion of Eq. (5.73), we indicate how the noise term $ \xi$(t) can be motivated by stochastic spike arrival.

Figure 5.12: Noisy integration. A stochastic contribution in the input current of an integrate-and-fire neuron causes the membrane potential to drift away from the reference trajectory (thick solid line). The neuron fires if the noisy trajectory (thin line) hits the threshold $ \vartheta$ (schematic figure).

5.5.1 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.

Figure 5.13: Each neuron receives input spikes from a large number of presynaptic neurons. Only a small portion of the input comes from neurons within the model network; other input is described as stochastic spike arrival.

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

$\displaystyle {\frac{{{\text{d}}}}{{{\text{d}}t}}}$u = - $\displaystyle {u\over \tau_m}$ + $\displaystyle {1\over C}$Iext(t) + $\displaystyle \sum_{j}^{}$$\displaystyle \sum_{{t_j^{(f)}>\hat{t}}}^{}$wj $\displaystyle \delta$(t - tj(f))  + $\displaystyle \sum_{k}^{}$$\displaystyle \sum_{{t_k^{(f)}>\hat{t}}}^{}$wk $\displaystyle \delta$(t - tk(f)) , (5.74)

where $ \delta$ is the Dirac $ \delta$ function and wj is the coupling to other presynaptic neurons j in the network. Input from background neurons is weighted by the factor wk. Firing times tk(f) of a background neuron k are generated by a Poisson process with mean rate $ \nu_{k}^{}$. Eq. (5.74) is called Stein's model (Stein, 1967b,1965).

In Stein's model, each input spike generates a postsynaptic potential $ \Delta$u(t) = wj$ \epsilon$(t - tj(f)) with $ \epsilon$(s) = e-s/$\scriptstyle \tau_{m}$ $ \Theta$(s), i.e., the potential jumps upon spike arrival by an amount wj 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 $ \epsilon$(s) that are generated by stochastic spike arrival; cf. Fig. 5.14A. Example: Membrane potential fluctuations

We consider stochastic spike arrival at rate $ \nu$. Each input spike evokes a postsynaptic potential w0 $ \epsilon$(s). The input statistics is assumed to be Poisson, i.e., firing times are independent. Thus, the input spike train,

S(t) = $\displaystyle \sum_{{k=1}}^{N}$$\displaystyle \sum_{{t_k^{(f)}}}^{}$$\displaystyle \delta$(t - tk(f)) , (5.75)

that arrives at neuron i is a random process with expectation

$\displaystyle \langle$S(t)$\displaystyle \rangle$ = $\displaystyle \nu_{0}^{}$ (5.76)

and autocorrelation

$\displaystyle \langle$S(tS(t')$\displaystyle \rangle$ -  $\displaystyle \nu_{0}^{2}$ = N $\displaystyle \nu_{0}^{}$ $\displaystyle \delta$(t - t') ; (5.77)

cf. Eq. (5.36).

Figure 5.14: Input spikes arrive stochastically (lower panel) at a mean rate of 1 kHz. A. Each input spike evokes an EPSP $ \epsilon$(s) $ \propto$ s exp(- s/$ \tau$) with $ \tau$ = 4 ms. The first EPSP (the one generated by the spike at t = 0) is plotted. The EPSPs of all spikes sum up and result in a fluctuating membrane potential u(t). B. Continuation of the simulation shown in A. The horizontal lines indicate the mean (dotted line) and the standard deviation (dashed lines) of the membrane potential.
{\bf A}
\end{minipage} \vspace{-35mm}

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

u(t) = w0$\displaystyle \int_{0}^{\infty}$$\displaystyle \epsilon_{0}^{}$(sS(t - s) ds . (5.78)

We are interested in the mean potential u0 = $ \langle$u(t)$ \rangle$ and the variance $ \langle$$ \Delta$u2$ \rangle$ = $ \langle$[u(t) - u0]2$ \rangle$. Using Eqs. (5.76) and (5.77) we find

u0 = w0 $\displaystyle \nu_{0}^{}$ $\displaystyle \int_{0}^{\infty}$$\displaystyle \epsilon_{0}^{}$(s) ds (5.79)


$\displaystyle \langle$$\displaystyle \Delta$u2$\displaystyle \rangle$ = w02 $\displaystyle \int_{0}^{\infty}$$\displaystyle \int_{0}^{\infty}$$\displaystyle \epsilon_{0}^{}$(s$\displaystyle \epsilon_{0}^{}$(s')  $\displaystyle \langle$S(tS(t')$\displaystyle \rangle$ ds ds'  - u02    
  = w02 $\displaystyle \nu_{0}^{}$$\displaystyle \int_{0}^{\infty}$$\displaystyle \epsilon_{0}^{2}$(s) ds . (5.80)

In Fig. 5.14 we have simulated a neuron which receives input from N = 100 background neurons with rate $ \nu_{j}^{}$ = 10 Hz. The total spike arrival rate is therefore $ \nu_{0}^{}$ = 1 kHz. Each spike evokes an EPSP w0 $ \epsilon$(s) = 0.1 (s/$ \tau$) exp(- s/$ \tau$) with $ \tau$ = 4ms. The evaluation of Eqs. (5.79) and (5.80) yields u0 = 0.4 and $ \sqrt{{\langle \Delta u^2 \rangle}}$ = 0.1.

Mean and fluctuations for Stein's model can be derived by evaluation of Eqs. (5.79) and (5.80) with $ \epsilon$(s) = e-s/$\scriptstyle \tau_{m}$. The result is

u0 = w0 $\displaystyle \nu_{0}^{}$ $\displaystyle \tau_{m}^{}$ (5.81)
$\displaystyle \langle$$\displaystyle \Delta$u2$\displaystyle \rangle$ = 0.5 w02 $\displaystyle \nu_{0}^{}$ $\displaystyle \tau_{m}^{}$ (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. Example: Balanced excitation and inhibition

Let us suppose that an integrate-and-fire neuron defined by Eq. (5.74) with $ \tau_{m}^{}$ = 10ms receives input from 100 excitatory neurons ( wk = + 0.1) and 100 inhibitory neurons ( wk = - 0.1). Each background neuron k fires at a rate of $ \nu_{k}^{}$ = 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 Iext = I0 > 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$ \ge$$ \vartheta$, the membrane potential is reset to ur = 0.

Figure 5.15: A. Voltage trajectory of an integrate-and-fire neuron ( $ \tau_{m}^{}$ = 10 ms, ur = 0) driven by stochastic excitatory and inhibitory spike input at $ \nu_{+}^{}$ = $ \nu_{-}^{}$ = 1 kHz. Each input spike causes a jump of the membrane potential by w± = ±0.1. The neuron is biased by a constant current I0 = 0.8 which drives the membrane potential to a value just below the threshold of $ \vartheta$ = 1 (horizontal line). Spikes are marked by vertical lines. B. Similar plot as in A except that the jumps are smaller ( w± = ±0.025) while rates are higher ( $ \nu_{{\pm}}^{}$ = 16 kHz).
{\bf A}

We note that the mean of the stochastic background input vanishes since $ \sum_{k}^{}$wk $ \nu_{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

$\displaystyle \langle$$\displaystyle \Delta$u2$\displaystyle \rangle$ = 0.5 $\displaystyle \tau_{m}^{}$$\displaystyle \sum_{k}^{}$wk2 $\displaystyle \nu_{k}^{}$ = 0.1 ; (5.83)

cf. Section 5.5.2 for a different derivation. Let us now increase all rates by a factor of a > 1 and multiply at the same time the synaptic efficacies by a factor 1/$ \sqrt{{a}}$. Then both mean and variance of the stochastic background input are the same as before, but the size wk of the jumps is decreased; cf. Fig. 5.15B. In the limit of a$ \to$$ \infty$ the jump process turns into a diffusion process and we arrive at the stochastic model of Eq. (5.73). A systematic discussion of the diffusion limit is the topic of the next subsection.

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 $ \hat{{t}}$. Immediately after the firing the membrane potential was reset to ur. Because of the stochastic spike arrival, we cannot predict the membrane potential for t > $ \hat{{t}}$, but we can calculate its probability density, p(u, t).

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

Prob$\displaystyle \big\{$no spike in [t, t + $\displaystyle \Delta$t]$\displaystyle \big\}$ = 1 - $\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(t$\displaystyle \Delta$t . (5.84)

If no spike arrives in [t, t + $ \Delta$t], the membrane potential changes from u(t) = u' to u(t + $ \Delta$t) = u' exp(- $ \Delta$t/$ \tau_{m}^{}$). On the other hand, if a spike arrives at synapse k, the membrane potential changes from u' to u' exp(- $ \Delta$t/$ \tau_{m}^{}$) + wk. Given a value of u' at time t, the probability density of finding a membrane potential u at time t + $ \Delta$t is therefore given by
Ptrans(u, t + $\displaystyle \Delta$t| u', t) = $\displaystyle \big[$1 - $\displaystyle \Delta$t$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(t)$\displaystyle \big]$ $\displaystyle \delta$$\displaystyle \left(\vphantom{u - u'\, e^{-\Delta t/\tau_m}}\right.$u - u' e-$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$$\displaystyle \left.\vphantom{u - u'\, e^{-\Delta t/\tau_m}}\right)$  
    + $\displaystyle \Delta$t$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(t)$\displaystyle \delta$$\displaystyle \left(\vphantom{u - u'\, e^{-\Delta t/\tau_m}- w_k}\right.$u - u' e-$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$ - wk$\displaystyle \left.\vphantom{u - u'\, e^{-\Delta t/\tau_m}- w_k}\right)$ . (5.85)

We will refer to Ptrans as the transition law. Since the membrane potential is given by the differential equation (5.74) with input spikes generated by a Poisson distribution, the evolution of the membrane potential is a Markov Process (i.e., a process without memory) and can be described by (van Kampen, 1992)

p(u, t + $\displaystyle \Delta$t) = $\displaystyle \int$Ptrans(u, t + $\displaystyle \Delta$t| u', tp(u', t) du' . (5.86)

We put Eq. (5.85) in (5.86). To perform the integration, we have to recall the rules for $ \delta$ functions, viz., $ \delta$(a u) = a-1 $ \delta$(u). The result of the integration is
p(u, t + $\displaystyle \Delta$t) = $\displaystyle \big[$1 - $\displaystyle \Delta$t$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(t)$\displaystyle \big]$ e$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$ p$\displaystyle \left(\vphantom{e^{\Delta t/\tau_m}\,u,t}\right.$e$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$ u, t$\displaystyle \left.\vphantom{e^{\Delta t/\tau_m}\,u,t}\right)$  
    + $\displaystyle \Delta$t$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(te$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$ p$\displaystyle \left(\vphantom{e^{\Delta t/\tau_m}\,u - w_k ,t}\right.$e$\scriptstyle \Delta$t/$\scriptstyle \tau_{m}$ u - wk, t$\displaystyle \left.\vphantom{e^{\Delta t/\tau_m}\,u - w_k ,t}\right)$ . (5.87)

Since $ \Delta$t is assumed to be small, we expand Eq. (5.87) about $ \Delta$t = 0 and find to first order in $ \Delta$t
$\displaystyle {p(u,t+\Delta t) - p(u,t) \over \Delta t}$ = $\displaystyle {1\over \tau_m}$ p(u, t) + $\displaystyle {1\over \tau_m}$ u $\displaystyle {\partial \over \partial u}$p(u, t)  
    + $\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(t$\displaystyle \left[\vphantom{ p(u-w_k,t) - p(u,t) }\right.$p(u - wk, t) - p(u, t)$\displaystyle \left.\vphantom{ p(u-w_k,t) - p(u,t) }\right]$ . (5.88)

For $ \Delta$t$ \to$ 0, the left-hand side of Eq. (5.88) turns into a partial derivative $ \partial$p(u, t)/$ \partial$t. Furthermore, if the jump amplitudes wk are small, we can expand the right-hand side of Eq. (5.88) with respect to u about p(u, t):
$\displaystyle \tau_{m}^{}$$\displaystyle {\partial \over \partial t}$p(u, t) = - $\displaystyle {\partial \over \partial u}$$\displaystyle \left[\vphantom{ -u + \tau_m \sum_k \nu_k(t) \, w_k}\right.$ - u + $\displaystyle \tau_{m}^{}$$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(twk$\displaystyle \left.\vphantom{ -u + \tau_m \sum_k \nu_k(t) \, w_k}\right]$ p(u, t)  
    + $\displaystyle {1\over 2}$ $\displaystyle \left[\vphantom{\tau_m \sum_k \nu_k(t) \, w_k^2}\right.$$\displaystyle \tau_{m}^{}$$\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(twk2$\displaystyle \left.\vphantom{\tau_m \sum_k \nu_k(t) \, w_k^2}\right]$ $\displaystyle {\partial^2\over \partial u^2}$p(u, t) (5.89)

where we have neglected terms of order wk3 and higher. The expansion in wk is called the Kramers-Moyal expansion. Eq. (5.89) is an example of a Fokker-Planck equation (van Kampen, 1992), i.e., a partial differential equation that describes the temporal evolution of a probability distribution. The right-hand side of Eq. (5.89) has a clear interpretation: The first term in rectangular brackets describes the systematic drift of the membrane potential due to leakage ( $ \propto$ - u) and mean background input ( $ \propto$ $ \sum_{k}^{}$$ \nu_{k}^{}$(twk). The second term in rectangular brackets corresponds to a `diffusion constant' and accounts for the fluctuations of the membrane potential. The Fokker-Planck Eq. (5.89) is equivalent to the Langevin equation (5.73) with R I(t) = $ \tau_{m}^{}$$ \sum_{k}^{}$$ \nu_{k}^{}$(twk and time-dependent noise amplitude

$\displaystyle \sigma^{{2}}_{{}}$(t) =  $\displaystyle \tau_{m}^{}$ $\displaystyle \sum_{k}^{}$$\displaystyle \nu_{k}^{}$(twk2 . (5.90)

The specific process generated by the Langevin-equation (5.73) with constant noise amplitude $ \sigma$ is called the Ornstein-Uhlenbeck process (Uhlenbeck and Ornstein, 1930).

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

$\displaystyle \sum_{{n=3}}^{\infty}$$\displaystyle {(-1)^n \over n!}$ An(t)$\displaystyle {\partial^n \over \partial u^n}$p(u, t) (5.91)

with An = $ \sum_{k}^{}$$ \nu_{k}^{}$(twkn. What are the conditions that these terms vanish? As in the example of Figs. 5.15A and B, we consider a sequence of models where the size of the weights wk decreases so that An$ \to$ 0 for n$ \ge$3 while the mean $ \sum_{k}^{}$$ \nu_{k}^{}$(twk and the second moment $ \sum_{k}^{}$$ \nu_{k}^{}$(twk2 remain constant. It turns out, that, given both excitatory and inhibitory input, it is always possible to find an appropriate sequence of models (Lansky, 1997,1984). For wk$ \to$ 0, the diffusion limit is attained and Eq. (5.89) is exact. For excitatory input alone, however, such a sequence of models does not exist (Plesser, 1999).

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

p($\displaystyle \vartheta$, t) $\displaystyle \equiv$ 0     for all t . (5.92)

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. Example: Free distribution

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

u(t|$\displaystyle \hat{{t}}$) = $\displaystyle {R \over \tau_m}$$\displaystyle \int_{0}^{{t-\hat{t}}}$e-s/$\scriptstyle \tau_{m}$ I(t - s) ds  + $\displaystyle {1 \over {\tau_m}}$$\displaystyle \int_{{0}}^{{t-\hat{t}}}$ e-s/$\scriptstyle \tau_{m}$ $\displaystyle \xi$(t - s) ds (5.93)

Since $ \langle$$ \xi$(t)$ \rangle$ = 0, the expected trajectory of the membrane potential is

u0(t) = $\displaystyle \langle$u(t|$\displaystyle \hat{{t}}$)$\displaystyle \rangle$ = $\displaystyle {R \over \tau_m}$$\displaystyle \int_{0}^{{t-\hat{t}}}$e-s/$\scriptstyle \tau_{m}$ I(t - s) ds . (5.94)

In particular, for constant input current I(t) $ \equiv$ I0 we have

u0(t) = u$\scriptstyle \infty$ $\displaystyle \left[\vphantom{ 1 - e^{-(t-\hat{t})/\tau_m} }\right.$1 - e-(t-$\scriptstyle \hat{{t}}$)/$\scriptstyle \tau_{m}$$\displaystyle \left.\vphantom{ 1 - e^{-(t-\hat{t})/\tau_m} }\right]$  (5.95)

with u$\scriptstyle \infty$ = R I0. Note that the expected trajectory is that of the noiseless model.

The fluctuations of the membrane potential have variance $ \langle$$ \Delta$u2$ \rangle$ = $ \langle$[u(t|$ \hat{{t}}$) - u0(t)]2$ \rangle$ with u0(t) given by Eq. (5.94). The variance can be evaluated with the help of Eq. (5.93), i.e.,

$\displaystyle \langle$$\displaystyle \Delta$u2(t)$\displaystyle \rangle$ = $\displaystyle {1 \over \tau_m^2}$$\displaystyle \int_{0}^{{t-\hat{t}}}$ds$\displaystyle \int_{0}^{{t-\hat{t}}}$ds' e-s/$\scriptstyle \tau_{m}$ e-s'/$\scriptstyle \tau_{m}$$\displaystyle \langle$$\displaystyle \xi$(t - s$\displaystyle \xi$(t - s')$\displaystyle \rangle$ . (5.96)

We use $ \langle$$ \xi$(t - s$ \xi$(t - s')$ \rangle$ = $ \sigma^{2}_{}$ $ \tau_{m}^{}$ $ \delta$(s - s') and perform the integration. The result is

$\displaystyle \langle$$\displaystyle \Delta$u2(t)$\displaystyle \rangle$ = $\displaystyle {1\over 2}$ $\displaystyle \sigma^{2}_{}$ $\displaystyle \left[\vphantom{ 1 - e^{-2(t-\hat{t})/\tau_m} }\right.$1 - e-2(t-$\scriptstyle \hat{{t}}$)/$\scriptstyle \tau_{m}$$\displaystyle \left.\vphantom{ 1 - e^{-2(t-\hat{t})/\tau_m} }\right]$ . (5.97)

Hence, noise causes the actual membrane trajectory to drift away from the noiseless reference trajectory u0(t). The typical distance between the actual trajectory and the mean trajectory approaches with time constant $ \tau_{m}^{}$/2 a limiting value

$\displaystyle \sqrt{{\langle \Delta u^2_\infty \rangle}}$ = $\displaystyle {1\over \sqrt{2}}$ $\displaystyle \sigma$/TD> (5.98)

Figure 5.16: In the absence of a threshold the membrane potential follows a Gaussian distribution around the noise-free reference trajectory u0(t) (schematic figure).

The solution of the Fokker-Planck equation (5.89) with initial condition p(u,$ \hat{{t}}$) = $ \delta$(u - ur) is a Gaussian with mean u0(t) and variance $ \langle$$ \Delta$u2(t)$ \rangle$, i.e.,

p(u, t) = $\displaystyle {1\over \sqrt{2\pi\,\langle \Delta u^2(t) \rangle}}$ exp$\displaystyle \left\{\vphantom{ - {[u(t\vert\hat{t}) - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle} }\right.$ - $\displaystyle {[u(t\vert\hat{t}) - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle}$$\displaystyle \left.\vphantom{ - {[u(t\vert\hat{t}) - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle} }\right\}$ (5.99)

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 t$ \to$$ \infty$ for constant input I0 is

p(u,$\displaystyle \infty$) = $\displaystyle {1\over \sqrt{\pi}}$$\displaystyle {1\over \sigma}$exp$\displaystyle \left\{\vphantom{ {[u-R\,I_0]^2 \over \sigma^2} }\right.$$\displaystyle {[u-R\,I_0]^2 \over \sigma^2}$$\displaystyle \left.\vphantom{ {[u-R\,I_0]^2 \over \sigma^2} }\right\}$ , (5.100)

which describes a Gaussian distribution with mean u$\scriptstyle \infty$ = R I0 and variance $ \sigma$/$ \sqrt{{2}}$.

5.5.3 Interval distribution

Let us consider a neuron that starts at time $ \hat{{t}}$ with a membrane potential ur and is driven for t > $ \hat{{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 > $ \hat{{t}}$, but only the probability that the membrane potential is in a certain interval [u0, u1]. Specifically, we have

Prob$\displaystyle \left\{\vphantom{ u_0 \!<\! u(t) \!<\! u_1\,\vert\,u(\hat{t})\!=\!u_r}\right.$u0 < u(t) < u1 | u($\displaystyle \hat{{t}}$) = ur$\displaystyle \left.\vphantom{ u_0 \!<\! u(t) \!<\! u_1\,\vert\,u(\hat{t})\!=\!u_r}\right\}$  =  $\displaystyle \int_{{u_0}}^{{u_1}}$p(u, t) du (5.101)

where p(u, t) is the distribution of the membrane potential at time t. In the diffusion limit, p(u, t) can be found by solution of the Fokker-Planck equation Eq. (5.89) with initial condition p(u,$ \hat{{t}}$) = $ \delta$(u - ur) and boundary condition p($ \vartheta$, t) = 0. At any time t > $ \hat{{t}}$, the survivor function,

SI(t|$\displaystyle \hat{{t}}$) = $\displaystyle \int_{{-\infty}}^{\vartheta}$p(u, t) du , (5.102)

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

PI(t|$\displaystyle \hat{{t}}$) = - $\displaystyle {\frac{{{\text{d}}}}{{{\text{d}}t}}}$$\displaystyle \int_{{-\infty}}^{\vartheta}$p(u, t) du . (5.103)

We recall that PI(t|$ \hat{{t}}$$ \Delta$t for $ \Delta$t$ \to$ 0 is the probability that a neuron fires its next spike between t and t + $ \Delta$t given a spike at $ \hat{{t}}$ and input I. In the context of noisy integrate-and-fire neurons PI(t|$ \hat{{t}}$) is called the distribution of `first passage times'. The name is motivated by the fact, that firing occurs when the membrane potential crosses $ \vartheta$ for the first time. Unfortunately, no general solution is known for the first passage time problem of the Ornstein-Uhlenbeck process. For constant input I(t) = I0, however, it is at least possible to give a moment expansion of the first passage time distribution. In particular, the mean of the first passage time can be calculated in closed form. Example: Mean interval for constant input

For constant input I0 the mean interspike interval is $ \langle$s$ \rangle$ = $ \int_{0}^{\infty}$s PI0(s| 0)ds = $ \int_{0}^{\infty}$s P0(s)ds; cf. Eq. (5.21). For the diffusion model Eq. (5.73) with threshold $ \vartheta$ reset potential ur, and membrane time constant $ \tau_{m}^{}$, the mean interval is

$\displaystyle \langle$s$\displaystyle \rangle$ = $\displaystyle \tau_{m}^{}$ $\displaystyle \sqrt{{\pi}}$$\displaystyle \int_{{{u_r - h_0 \over \sigma}}}^{{{\vartheta - h_0 \over \sigma}}}$du exp$\displaystyle \left(\vphantom{u^2}\right.$u2$\displaystyle \left.\vphantom{u^2}\right)$ $\displaystyle \left[\vphantom{1+ {\rm erf}(u)}\right.$1 + erf(u)$\displaystyle \left.\vphantom{1+ {\rm erf}(u)}\right]$ , (5.104)

where h0 = R I0 is the input potential caused by the constant current I0 (Johannesma, 1968). This expression can be derived by several methods; for reviews see, e.g., (Tuckwell, 1988; van Kampen, 1992). We will return to Eq. (5.104) in Chapter 6.2.1 in the context of populations of spiking neurons. Example: Numerical evaluation of PI(t|$ \hat{{t}}$)

Figure 5.17: Without a threshold, several trajectories can reach at time t the same value u = $ \vartheta$ from above or below.

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

Ptrans(u, t| u', t') = $\displaystyle {1\over \sqrt{2\pi\,\langle \Delta u^2(t) \rangle}}$ exp$\displaystyle \left\{\vphantom{ - {[u - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle} }\right.$ - $\displaystyle {[u - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle}$$\displaystyle \left.\vphantom{ - {[u - u_0(t)]^2 \over 2 \,\langle \Delta u^2(t) \rangle} }\right\}$ (5.105)

u0(t) = u' e-(t-t')/$\scriptstyle \tau_{m}$ + $\displaystyle \int_{0}^{{t-t'}}$e-s'/$\scriptstyle \tau_{m}$ I(t - s') ds (5.106)
$\displaystyle \langle$$\displaystyle \Delta$u2(t)$\displaystyle \rangle$ = $\displaystyle {\sigma^2\over 2}$$\displaystyle \left[\vphantom{ 1 - e^{-2\,(t-s)/\tau_m} }\right.$1 - e-2 (t-s)/$\scriptstyle \tau_{m}$$\displaystyle \left.\vphantom{ 1 - e^{-2\,(t-s)/\tau_m} }\right]$ . (5.107)

A method due to Schrödinger uses the solution of the unbounded problem in order to calculate the input-dependent interval distribution PI(t|$ \hat{{t}}$) of the diffusion model with threshold (Schrödinger, 1915; Plesser and Tanaka, 1997; Burkitt and Clark, 1999). The idea of the solution method is illustrated in Fig. 5.17. Because of the Markov property, the probability density to cross the threshold (not necessarily for the first time) at a time t, is equal to the probability to cross it for the first time at t' < t and to return back to $ \vartheta$ at time t, that is,

Ptrans($\displaystyle \vartheta$, t| ur,$\displaystyle \hat{{t}}$) = $\displaystyle \int_{{\hat{t}}}^{t}$PI(t'|$\displaystyle \hat{{t}}$Ptrans($\displaystyle \vartheta$, t|$\displaystyle \vartheta$, t') dt' . (5.108)

This integral equation can be solved numerically for the distribution PI(t'|$ \hat{{t}}$) for arbitrary input current I(t) (Plesser, 2000). An example is shown in Fig. 5.18.

Figure 5.18: A time-dependent input current I(t) generates a noise-free membrane potential u0(t) shown in the lower part of the figure. In the presence of diffusive noise, spikes can be triggered although the reference trajectory stays below the threshold (dashed line). This gives rise to an input-dependent interval distribution PI(t| 0) shown in the upper panel. Taken from (Plesser and Gerstner, 2000).
\centerline{\includegraphics[width=80mm]{Plesser2a-single.eps}} \centerline{\includegraphics[width=80mm]{Plesser2b.eps}}\par\vspace{0mm}

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Next: 5.6 The subthreshold regime Up: 5. Noise in Spiking Previous: 5.4 Slow noise in
Gerstner and Kistler
Spiking Neuron Models. Single Neurons, Populations, Plasticity
Cambridge University Press, 2002

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