Contents - Max-Planck-Institut für Physik komplexer Systeme

Main results.
probability density [kHz]
1. In the case of deterministic adaptation we find
that the ISI density can be well approximated
by an inverse Gaussian probability density. This
density is equal to the first-passage-time density
of a Brownian motion with a drift corresponding
to a base current that is reduced by the adaptation
(Fig. 2A). Furthermore, we have derived an
analytical formula for the serial correlation coefficient
ρn, which shows that neighboring ISIs are
negatively correlated (Fig. 3). This characteristic
feature has been numerically found in other
neuron models with deterministic adaptation and
fast fluctuations (e.g. [2, 3]) and has been experimentally
observed for a variety of neurons (for
reviews see [4, 5]).
2. A stochastic adaptation current can be approximated
by a long-correlated colored noise process
with effective parameters (reduced values
of mean, noise variance, correlation time). As a
consequence, the ISI densities become strongly
skewed and peaked compared to an inverse
Gaussian distribution (Fig. 2B) Another striking
difference is, that the ISI correlations are positive
and not negative as in the case of fast noise
(Fig. 3). We have provided analytical formulas for
the skewness, kurtosis and serial correlation coefficient
of the ISIs that clearly explain these properties.
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A
simulation
inverse Gaussian
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interspike interval [ms]
B
channel model
theory
inv. Gaussian
Figure 2: (A) Interspike interval (ISI) density for the case of deterministic
adaptation. The gray bars display the result from a simulation of
the model, the dashed line represents the inverse Gaussian distribution
obtained from the theory. (B) ISI density in the case of stochastic
adaptation. An inverse Gaussian distribution (dashed line) with the
same mean ISI and the same variance does not fit the ISI density obtained
from the simulation of the model with adaptation channels
(gray bars). Circles depict simulation results of the diffusion approximation
(Gaussian approximation) of the channel noise, the solid line
shows our theoretical approximation.
Conclusions. We have put forward several novel analytical
results that characterize the non-renewal spiking
behavior of neurons with noise and stochastic or
deterministic adaptation. Our analysis has revealed
a striking difference of the ISI statistics depending on
whether slow adaptation noise or fast current noise is
dominating. These findings suggest an indirect way to
determine the dominant source of noise on the basis
of the ISI statistics. As an application, we have used
our theoretical results to understand the origin of neuronal
response variability of auditory receptor cells of
locusts (collaboration with J. Benda, Munich). Preliminary
analysis indicates that slow adaptation currents
may indeed contribute to neuronal variability of this
primary sensory neuron.
serial correlation coefficient
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determ. adaptation
stoch. adaptation
theory
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lag
Figure 3: Serial correlation coefficient ρn as a function of the lag n.
In the case of deterministic adaptation interspike intervals (ISIs) are
negatively correlated, whereas in the case of stochastic adaptation
ISIs become positively correlated.
[1] T. Schwalger, K. Fisch, J. Benda, and B. Lindner. PLoS Comp.
Biol., 6(12):e1001026, 2010.
[2] M. J. Chacron, K. Pakdaman, and A. Longtin. Neural Comp.,
15:253, 2003.
[3] Y.-H. Liu and X.-J. Wang. J. Comp. Neurosci., 10:25, 2001.
[4] O. Avila-Akerberg and M. J. Chacron. Experimental Brain Research,
2011.
[5] F. Farkhooi, M. F. Strube-Bloss, and M. P. Nawrot. Phys. Rev. E,
79(2):021905–10, 2009.
2.9. How Stochastic Adaptation Currents Shape Neuronal Firing Statistics 59