5.4 Slow noise in the parameters

In one of the previous examples (`Motivating sigmoidal escape rates' in
Section 5.3.1), a new value of the threshold was chosen
at each time step; cf. Eq. (5.49). If time steps are
short enough, such an approach is closely related to escape rate models. A
completely different class of noise models can be constructed if the value of
a parameter is changed *after each spike*. Thus between two spikes the
noise is `frozen' so that the value of the fluctuating parameter does not
change. In other words, the noise is slow compared to the fast neuronal
dynamics. In principle, any of the neuronal parameters such as threshold,
membrane time constant, or length of the refractory period, can be subject
to this type of noise (Gerstner, 2000b; Lansky and Smith, 1989; Gestri, 1978; Knight, 1972a).
In this section
we want to show how to analyze such slow variations and calculate the interval
distribution. We emphasize that these `slow' noise models cannot be mapped
onto an escape rate formalism.

To keep the arguments simple, we will concentrate on noise in the formulation of reset and refractoriness. We assume an exponential refractory kernel,

with time constant . In order to introduce noise, we suppose that the amount of the reset depends on a stochastic variable

where < 0 is a fixed parameter and

In the `noisy reset' model, firing is given by the threshold condition

= u(t|, r) = (t - ) + (t - , s) I(t - s) ds , |
(5.66) |

where (

T(, r) = mint - | u(t|, r) = . |
(5.67) |

If

Let us now evaluate the interval distribution (5.68) for the variant
SRM_{0} of the Spike Response Model,

u(t|, r) = (t - ) + h(t) , |
(5.69) |

with constant input potential

(t - ) = exp[- (t - - r)/] , |
(5.70) |

which is identical to that of a noiseless neuron that has fired its last spike at

Thus, the Gaussian distribution

Even though stochastic reset is not a realistic noise model for individual
neurons, noise in the parameter values can approximate *inhomogeneous*
populations of neurons where parameters vary
from one neuron to the next (Wilson and Cowan, 1972; Knight, 1972a). Similarly, a
fluctuating background input that changes slowly compared to the typical
interspike interval can be considered as a slow change in the value of the
firing threshold. More generally, noise with a cut-off frequency smaller than
the typical firing rate can be described as slow noise in the parameters.

Cambridge University Press, 2002

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