Subsections

• 7.3.1 Signal Term
  • 7.3.1.1 Example: Slow noise in the parameters
  • 7.3.1.2 Example: Escape Noise
  • 7.3.1.3 Example: Diffusive noise (*)
  
  
• 7.3.2 Signal-to-Noise Ratio

7.3 Transfer Function

Our considerations regarding step current input can be generalized to an arbitrary input current I(t) that is fluctuating around a mean value of I₀. We study a population of independent integrate-and-fire or SRM₀ neurons. The input current I(t) = I₀ + $\Delta$I(t) generates an input potential

$$\int_{0}^{\infty} \kappa_{0}^{} \Delta \Delta$$ (7.51)

where h₀ = R I₀ with R = $\int_{0}^{\infty}$$\kappa_{0}^{}$(s)ds is the mean input potential. In particular we want to know how well a periodic input current

$$\omega$$ (7.52)

can be transmitted by a population of neurons. The signal transfer function calculated in Section 7.3.1 characterizes the signal transmission properties as a function of the frequency $\omega$. The signal-to-noise ratio is the topic of Section 7.3.2.

7.3.1 Signal Term

We assume that the population is close to a state of asynchronous firing, viz., A(t) = A₀ + $\Delta$A(t). The linear response of the population to the change in the input potential h is given by Eq. (7.3) which can be solved for $\Delta$A by taking the Fourier transform. For $\omega$$\ne$ 0 we find

$$\hat{{A}} \omega \omega {A_0 \, \hat{{\mathcal{L}}}(\omega) \, \hat{\kappa } (\omega) \over 1 - \hat{P}(\omega)\,} \hat{{I}} \omega \hat{{G}} \omega \hat{{I}} \omega$$ (7.53)

Hats denote transformed quantities, i.e., $\hat{{\kappa }}$($\omega$) = $\int$$\kappa_{0}^{}$(s) exp(- i $\omega$ s) ds is the Fourier transform of the response kernel; $\hat{{P}}$($\omega$) is the Fourier transform of the interval distribution; and $\hat{{{\mathcal{L}}}}$($\omega$) is the transform of the kernel $\mathcal {L}$. Note that for $\omega$$\ne$ 0 we have A($\omega$) = $\Delta$A($\omega$) and I($\omega$) = $\Delta$I($\omega$) since A₀ and I₀ are constant.

The function $\hat{{G}}$($\omega$), defined by Eq. (7.57), describes the (linear) response $\hat{{A}}$($\omega$) of a population of spiking neurons to a periodic signal $\hat{{I}}$($\omega$). It is also called the (frequency-dependent) gain of the system. Inverse Fourier transform of Eq. (7.57) yields

$$\int_{0}^{\infty} \Delta$$ (7.54)

with

$${1\over 2\pi} \int_{{-\infty}}^{\infty} \hat{{G}} \omega \omega \omega$$ (7.55)

A₀ is the mean rate for constant drive I₀. Equation (7.58) allows us to calculate the linear response of the population to an arbitrary input current.

We can compare the amplitude of an input current at frequency $\omega$ with the amplitude of the response. The ratio

$$\hat{{G}} \omega {\hat{A}(\omega) \over \hat{I}(\omega)}$$ (7.56)

as a function of $\omega$ characterizes the signal transmission properties of the system. If |$\hat{{G}}$| decays for $\omega$ > $\omega_{0}^{}$ to zero, we say that $\hat{{G}}$ has a cut-off frequency $\omega_{0}^{}$. In this case, signal transmission at frequencies $\omega$ $\gg$ $\omega_{0}^{}$ is difficult. On the other hand, if |$\hat{{G}}$| approaches a positive value for $\omega$$\to$$\infty$, signal transmission is possible even at very high frequencies.

In the following examples signal we will study transmission properties of a population of neurons with different noise models. In particular, we will see that for slow noise in the parameters (e.g. noise in the reset) signal transmission is possible at very high frequencies (that is, there is no cut-off frequency) (Gerstner, 2000b; Knight, 1972a). On the other hand, for escape noise models the cut-off frequency depends on the noise level. For a large amount of escape noise, the cut-off frequency is given by the the inverse of the membrane time constant (Gerstner, 2000b). Finally, diffusive noise models have a cut-off frequency if the noise input is white (standard diffusion model), but do not have a cut-off frequency if the noise has a long correlation time (Brunel et al., 2001).

Even if there is no cut-off frequency for the transmission of fast input currents, we may not conclude that real neurons are infinitely fast. In fact, a finite time constant of synaptic channels leads to a frequency cut-off for the input current which may enter the cell. In this sense, it is the time constant of the synaptic current which determines the cut-off frequency of the population. The membrane time constant is of minor influence (Gerstner, 2000b; Treves, 1993; Knight, 1972a).

Figure 7.9: Signal gain for integrate-and-fire neurons with noisy reset (A) and escape noise (B). For low noise (short-dashed line) the variance of the interval distribution is $\sigma$ = 0.75ms; For high noise (long-dashed line) the variance is $\sigma$ = 4ms. Solid line: variance $\sigma$ = 2ms. Note that for noisy reset (slow noise) the signal transfer function has no cut-off frequency, whatever the noise level. The value of the bias current has been adjusted so that the mean interval is always 8ms. The escape rate in b is piecewise linear $\rho$ = $\rho_{0}^{}$ [u - $\vartheta$]$\mathcal {H}$(u - $\vartheta$); taken from Gerstner (2000b).

7.3.1.1 Example: Slow noise in the parameters

In this example, we consider integrate-and-fire neurons with noisy reset; cf. Chapter 5.4. For noisy reset the interval distribution in the stationary state is a Gaussian P₀(s) = $\mathcal {G}$_$\sigma$(s - T₀) with mean T₀ and width $\sigma$; cf. Eq. (7.38). Fourier transform of the interval distribution yields

$$\hat{{P}} \omega \left\{\vphantom{-{1\over 2}{\sigma}^2\omega^2 - i\omega T_0}\right. {1\over 2} \sigma^{{2}}_{{}} \omega^{2}_{} \omega \left.\vphantom{-{1\over 2}{\sigma}^2\omega^2 - i\omega T_0}\right\}$$ (7.57)

The kernel $\mathcal {L}$ may be read off from Eq. (7.41) or Tab. 7.1. Its Fourier transform is

$$\hat{{{\mathcal{L}}}} \omega {1\over u'} \left\{\vphantom{1 - \exp\left[-{1\over 2}{\sigma}^2\omega^2 - i \omega T_0 -{T_0\over\tau}\right]}\right. \left[\vphantom{-{1\over 2}{\sigma}^2\omega^2 - i \omega T_0 -{T_0\over\tau}}\right. {1\over 2} \sigma^{{2}}_{{}} \omega^{2}_{} \omega {T_0\over\tau} \left.\vphantom{-{1\over 2}{\sigma}^2\omega^2 - i \omega T_0 -{T_0\over\tau}}\right] \left.\vphantom{1 - \exp\left[-{1\over 2}{\sigma}^2\omega^2 - i \omega T_0 -{T_0\over\tau}\right]}\right\}$$ (7.58)

where u' is the slope of the noise-free membrane potential at the moment of threshold crossing.

We adjust the bias current I₀ so that the mean interspike interval of the neurons is T₀ = 8ms. In Fig. 7.9A we have plotted the gain |$\hat{{G}}$($\omega$)| = |$\hat{{A}}$($\omega$)/$\hat{{I}}$($\omega$)| as a function of the stimulation frequency f = $\omega$/(2$\pi$). For a medium noise level of $\sigma$ = 2ms, the signal gain has a single resonance at f = 1/T₀ = 125Hz. For lower noise, further resonances at multiples of 125 Hz appear. For a variant of the noisy reset model, a result closely related to Eq. (7.57) has been derived by Knight (1972a).

Independently of the noise level, we obtain for integrate-and-fire neurons for $\omega$$\to$ 0 the result |$\hat{{G}}$(0)| = J_extA₀[1 - exp(- T₀/$\tau$)]/(u' T₀). Most interesting is the behavior in the high-frequency limit. For $\omega$$\to$$\infty$ we find |$\hat{{G}}$($\omega$)|$\to$RA₀/(u' $\tau$), hence

$$\left\vert\vphantom{{\hat{G}(\infty) \over \hat{G}(0) }}\right. {\hat{G}(\infty) \over \hat{G}(0)} \left.\vphantom{{\hat{G}(\infty) \over \hat{G}(0) }}\right\vert {T_0\over\tau} \left[\vphantom{ 1 - e^{-T_0/\tau}}\right. \tau \left.\vphantom{ 1 - e^{-T_0/\tau}}\right]^{{-1}}_{}$$ (7.59)

We emphasize that the high-frequency components of the current are not attenuated by the population activity - despite the integration on the level of the individual neurons. The reason is that the threshold process acts like a differentiator and reverses the low-pass filtering of the integration. In fact, Eq. (7.63) shows that high frequencies can be transmitted more effectively than low frequencies. The good transmission characteristics of spiking neurons at high frequencies have been studied by Knight (1972a), Gerstner (2000b), and Brunel et al. (2001). They were also confirmed experimentally by Knight (1972b) and F. Chance (private communication).

So far we have discussed results of the linearized theory; viz., Eqs. (7.41) and (7.57). The behavior of the full non-linear system is shown in Fig. 7.10. A population of unconnected SRM₀ neurons is stimulated by a time-dependent input current which was generated as a superposition of 4 sinusoidal components with frequencies at 9, 47, 111 and 1000Hz which have been chosen arbitrarily. The activity equation A(t) = $\int_{{-\infty}}^{t}$P_I(t|$\hat{{t}}$) A($\hat{{t}}$) d$\hat{{t}}$ been integrated with time steps of 0.05ms and the results are compared with those of a simulation of a population of 4000 neurons. The 1kHz component of the signal I(t) is clearly reflected in the population A(t). Theory and simulation are in excellent agreement.

Figure 7.10: Response of the population activity (top) of SRM₀ neurons with noisy reset to a time dependent current (bottom). The current is a superposition of 4 sine waves at 9, 47, 111, and 1000Hz. The simulation of a population of 4000 neurons (solid line, top) is compared with the numerical integration (dashed line) of the population equation (6.75). Note that even the 1kHz component of the signal is well transmitted. Parameters: exponential response function with time constant $\tau$ = 4ms. Threshold is $\vartheta$ = - 0.135 so that the mean activity is A = 125Hz; noise $\sigma$ = 2ms; J₀ = 0; taken from (Gerstner, 2000b)

7.3.1.2 Example: Escape Noise

We have seen in the preceding section, that noisy reset is rather exceptional in the sense that the transient remains sharp even in the limit of high noise. To study the relevance of the noise model, we return to Eq. (7.57). The signal gain $\hat{{G}}$($\omega$) = | A($\omega$)/I($\omega$)| is proportional to $\hat{{{\mathcal{L}}}}$($\omega$). If the kernel $\mathcal {L}$(x) is broad, its Fourier transform $\hat{{{\mathcal{L}}}}$($\omega$) will fall off to zero at high frequencies and so does the signal gain $\hat{{G}}$($\omega$). In Fig. 7.9B we have plotted the signal gain $\hat{{G}}$($\omega$) for integrate-and-fire neurons with escape noise at different noise levels. At low noise, the result for escape noise is similar to that of reset noise (compare Figs. 7.9A and B) except for a drop of the gain at high frequencies. Increasing the noise level, however, lowers the signal gain of the system. For high noise (long-dashed line in Fig. 7.9B the signal gain at 1000 Hz is ten times lower than the gain at zero frequency. The cut-off frequency depends on the noise level. Note that for escape noise, the gain at zero frequency also changes with the level of noise.

7.3.1.3 Example: Diffusive noise (*)

It is possible to calculate the signal transmission properties of integrate-and-fire neurons with diffusive noise by a linearization of the population density equation (6.21) about the stationary membrane potential distribution p₀(u). The resulting formula (Brunel et al., 2001) is rather complicated but can be evaluated numerically. It is found that in the standard diffusion model the gain |$\hat{{G}}$($\omega$)| decays as 1/$\sqrt{{\omega}}$ for large $\omega$. Thus the gain exhibits a cut-off frequency similar to that found in the escape noise model.

Standard diffusive noise corresponds to a drive by stochastic $\delta$ current pulses, which is usually motivated as a description of stochastic spike arrival; cf. Chapter 5. In a more realistic model of stochastic spike arrival, input spikes evoke a current pulse of finite width. The duration of the current pulse is characterized by the synaptic time constant $\tau_{s}^{}$. In that case, the effective noisy input current has correlations on the time scale of $\tau_{s}^{}$. If $\tau_{s}^{}$ > 1/A₀, the noise is `slow' compared to the intrinsic firing rate of the neuron. It is found that with such a slow noise, the $\hat{{G}}$($\omega$) has no cut-off frequency (Brunel et al., 2001). In this limit, the gain factor is therefore similar to that of the stochastic reset model. In other words, we have the generic result that for `fast' noise the gain factor has a cut-off frequency whereas for `slow' noise it has not.

7.3.2 Signal-to-Noise Ratio

So far we have considered the signal transmission properties of a large population in the limit N$\to$$\infty$. In this case the population activity can be considered as a continuous signal, even though individual neurons emit short pulse-like action potentials. For a finite number N of neurons, however, the population activity A(t) will fluctuate around a time-dependent mean. In this section we want to estimate the amplitude of the fluctuations.

For independent neurons that are stimulated by a constant current I₀, we can calculate the noise spectrum of the population activity using the methods discussed in Chapter 5. In fact, the noise spectrum C_AA of the population activity is proportional to the Fourier transform of the autocorrelation function of a single-neuron:

$$\omega {1\over N} \omega$$ (7.60)

The proportionality factor takes care of the fact that the amplitude of the fluctuations of A(t) is inversely proportional to the number of neurons in the population. For constant input, we can calculate the single-neuron autocorrelation in the framework of stationary renewal theory. Its Fourier transform is given by Eq. (5.35) and is repeated here fore convenience:

$$\hat{{C}}_{{ii}}^{} \omega \nu_{i}^{} \left[\vphantom{ { 1 + \hat{P}_0(\omega) \over 1 - \hat{P}_0(\omega)} }\right. {1 + \hat{P}_0(\omega) \over 1 - \hat{P}_0(\omega)} \left.\vphantom{ { 1 + \hat{P}_0(\omega) \over 1 - \hat{P}_0(\omega)} }\right] \pi \nu_{i}^{2} \delta \omega$$ (7.61)

Here $\hat{{P}}_{0}^{}$($\omega$) is the Fourier transform of the inter-spike interval distribution in the stationary state.

If the amplitude of the periodic stimulation is small, the noise term of the population activity can be estimated from the stationary autocorrelation function. The signal-to-noise ratio at frequency $\omega$ is

$${\vert\hat{G}(\omega)\vert^2 \over \hat{C}_{AA}(\omega)} {\vert\hat{G}(\omega)\vert^2 \over \hat{C}_{ii}(\omega)}$$ (7.62)

where N is the number of neurons. The signal-to-noise ratio increases with N as expected.