Contents - Max-Planck-Institut für Physik komplexer Systeme

2.9 How Stochastic Adaptation Currents Shape Neuronal Firing Statistics
Channel noise and spike-frequency adaptation.
Neurons of sensory systems encode signals from the
environment by sequences of electrical pulses – socalled
spikes. This coding of information is fundamentally
limited by the presence of intrinsic neural noise.
One major noise source is channel noise that is generated
by the random activity of various types of ionic
channels in the cell membrane. In other words, the current
associated with a certain population of ion channels
exhibits fluctuations unless the population is infinitely
large. The fast currents that establish the spiking
mechanism can be thus a source of such channel
noise leading to a substantial variability of the neuronal
response.
Besides these fast spike-generating currents, slow
adaptation currents can also be a source of channel
noise. Adaptation currents mediate a long-term drop
of firing rate to a sustained stimulus, called spikefrequency
adaptation (SFA). SFA is found in many neurons
and profoundly shapes the signal transmission
properties of a neuron by emphasizing fast changes
in the stimulus but adapting the spiking frequency to
slow stimulus components. The role of such spikefrequency
adaptation for neural coding is, however,
still poorly understood in the context of stochastic
spike generation. In particular, the effects of different
kinds of noise such as fast channel noise and
slow adaptation channel noise on the neuronal spiking
statistics is still largely unknown.
Stochastic and deterministic adaptation. To gain a
better insight into the stochastic dynamics of adapting
neurons we have analyzed the spiking activity of
a noisy integrate-and-fire neuron model with an adaptation
current (Fig.1). In the model, neural noise originates
from either slow adaptation channel noise or
fast channel noise. Surprisingly, the two noise sources
lead to qualitatively different statistics of the interspike
intervals (ISI), i.e. the intervals between two
subsequent spikes [1]. The same difference in the ISI
statistics is also observed in simulations of a more detailed
Hodgkin-Huxley-type neuron model. These theoretical
findings suggest that higher-order ISI statistics
might help to experimentally distinguish adaptation
noise from fast channel noise as the dominating intrinsic
noise source of sensory neurons.
TILO SCHWALGER, BENJAMIN LINDNER
Figure 1: Illustration of the neuron model with a stochastic adaptation
current. The membrane potential V (bottom panel) is driven by
a constant base current, Gaussian white noise (modelling fast channel
noise) and an inhibitory adaptation current. Whenever the membrane
potential hits the threshold at V = 1 it is reset to the reset
potential Vreset = 0. These threshold events define the spike times of
the model. At the same time, adaptation channels can open during a
short time window after each spiking event increasing or decreasing
the open probability towards the steady state activation indicated in
the middle panel. As a result the fraction of open channels, and thus
the adaptation current, makes a random walk that increases immediately
after each spike and decays in between spikes (top panel). This
spike-triggered adaptation current inhibits in turn the dynamics of V
realizing a negative feedback mechanism.
For the theoretical analysis we considered a perfect
integrate-and-fire model with a voltage-gated adaptation
current (similar to the M-type current) for two limit
cases [1]:
1. Deterministic adaptation. The slow adaptation current
is taken to be deterministic corresponding to
a large (macroscopic) population of independent
adaptation channels. The variability of the ISIs is
solely caused by fast noise sources modeled by a
white Gaussian noise.
2. Stochastic adaptation. The ISI variability is only
due to the stochasticity of the adaptation current
resulting from a finite number of slow adaptation
channels.
From a mathematical point of view, both cases belong
to the class of non-renewal models, because the
slow adaptation current introduces memory of previous
ISIs. Such processes are notoriously difficult to analyze.
The non-renewal properties can be characterized
by the serial correlation coefficient ρn of the ISI
sequence (...,Ti−1,Ti,Ti+1,...) defined by
ρn = 〈TiTi+n〉 − 〈Ti〉 2
〈T 2 i 〉 − 〈Ti〉 2 . (1)
Hence, ρn measures the correlation between ISIs that
are lagged by n.
58 Selection of Research Results