Figure 1.
Integrate-and-fire dynamics with adaptation channels.
A Channel model: a population of independent voltage-gated ion channels, which can be either in an open or a closed state, mediate an adaptation current through a neuron's membrane. B PIF model: Subthreshold dynamics of the membrane potential
(bottom). The variable
(measured in units of
) is reset to a value
after crossing the threshold at
. Action potentials are not generated explicitely. Instead, the effect of an action potential is captured by the activation function
, which is set to one in a short time window of
following each threshold crossing of the IF model (middle panel). The adaptation current is proportional to the fraction of open channels
(top panel). The sample traces were obtained from a simulation of Eq. (35) with
channels, white noise intensity
, adaptation time constant
, base current
and maximal adaptation current
. C The time-dependent firing rate (top) in response to a step stimulus (bottom) is independent of the source of noise (stochastic adaptation – solid line, deterministic adaptation plus white noise – dashed line). The gray line shows the theory given by Eq. (55).
Figure 2.
Diffusion approximation of adaptation current.
A Sample traces of the integrate-and-fire dynamics with two-state adaptation channels, Eq. (20), (35) ( and
). The fraction of open channels
(top) exhibits discontinuous jumps with directions that depend on the presence or absence of a spike as given by the activation function
(middle panel). B Sample traces of the diffusion model, Eq. (1), with the same 1st and 2nd infinitesimal jump moments of
as in the channel model (A). C The fraction of open channels
can be split into the deterministic part
, Eq. (1b), corresponding to
, and an Ornstein-Uhlenbeck process
, Eq. (1c), with a correlation time equal to the adaptation time constant (colored noise). The parameters are
,
,
.
Figure 3.
ISI histograms of a PIF neuron – theory vs. simulation.
A ISI densities in the case of deterministic adaptation () for different noise intensities
. Gray bars show the histograms obtained from simulations of Eq. (2); solid lines display the mean-adaptation approximation, Eq. (64) (inverse Gaussian density). B ISI densities in the case of stochastic adaptation (
) for different
as indicated in the panels. The adaptation current was modeled either by the channel model (gray bars), Eq. (20), or by the diffusion model (circles), Eq. (1). The theory, Eq. (69), is displayed as a solid line. Parameters are chosen as in Fig. 2.
Figure 4.
Comparison of ISIHs for deterministic vs. stochastic adaptation.
A and C – The ISIH obtained from a simulation of the deterministic adaptation model, Eq. (2), with noise intensity can be well described by an inverse Gaussian distribution (dashed line), Eq. (64). B and D – ISIH for the stochastic adaptation model with
and
. The channel model (gray bars) is more peaked than an inverse Gaussian distribution, Eq. (64), with the same mean and CV (dashed line). The ISIH of the diffusion model, (simulation of Eq. (3), circles) is well described by the colored noise approximation, Eq. (69), (solid line). Note the double logarithmic axis in C and D revealing the tail of the distribution. Other parameters as in Fig. 2.
Figure 5.
Shape parameters of the ISIH for deterministic and stochastic adaptation.
A Rescaled skewness for deterministic adaptation (white squares) and for stochastic adaptation (channel model – white circles, diffusion model – black circles, colored noise approximation – gray circles). Different CVs were obtained by varying
or
. The dashed line depicts the theoretical curve, Eq. (7), and the solid line depicts the semi-analytical result obtained from the moments of the ISI density, Eq. (69), using numerical integration. B The corresponding plot for the rescaled kurtosis
. The adaptation time constant was
. All other parameters as in Fig. 2.
Figure 6.
Comparison of diffusion and channel model.
A The coefficient of variation as a function of the number of adaptation channels for the diffusion model (black circles, Eq. (1)), the channel model (white circles, Eq. (20)) and the colored noise approximation (grey circles, Eq. (4)). The dashed line depicts the theoretical curve Eq. (71) and the solid line depicts the semi-analytical result obtained from the moments of the ISI density, Eq. (69), using numerical integration. B The corresponding curves for the rescaled kurtosis
. The dashed line represents the theory given by Eq. (114). The time scale separation was
. Parameters as in Fig. 2.
Figure 7.
Shape parameters of the ISIH as a function of the time scale separation.
A Rescaled skewness and B rescaled kurtosis
for
channels (corresponding
) for stochastic adaptation (circles) and
for deterministic adaptation (squares). Theory Eq. (113) and (114) is displayed by the solid line, the line
is indicated by a dotted line. C and D corresponding plots for
channels (corresponding to
) and
.
was varied by changing
at a fixed
(
), all other parameters as in Fig. 2.
Figure 8.
Serial correlation coefficient as a function of the lag between ISIs.
A The case of deterministic adaptation with for different values of the time constant
(as indicated in the legend). The theoretical curves, Eq. (9), are depicted by solid lines; the zero baseline is indicated by a dotted line. B The case of stochastic adaptation with
for different values of the time constant
(as in A). The channel model, Eq. (20), is represented by white symbols, the diffusion approximation (Eq. (1)) is represented by black symbols. The theory based on the colored noise approximation, Eq. (4), is depicted by a solid line. Other parameters as in Fig. 2.
Figure 9.
Serial correlation coefficient at lag 1 as a function of the time scale separation.
A Serial correlation coefficient in the case of
channels (corresponding to
) for stochastic adaptation (circles, Eq. (3)) and
for deterministic adaptation (squares, Eq. (2)). Theoretical curves for stochastic adaptation, Eq. (72), and deterministic adaptation, Eq. (9), are displayed by a dashed line and a solid line, respectively. The zero baseline is indicated by a dotted line. B shows the corresponding plot for
channels (corresponding to
) and
. The gray-shaded region marks the relevant range for spike-frequency adaptation.
was varied by changing
at fixed
(
), all other parameters as in Fig. 2.
Figure 10.
ISI statistics of the PIF model in the presence of both stochastic adaptation and white noise.
For a fixed level of white noise () the number of adaptation channels
was varied. For a small channel population slow channel noise dominates (white region), whereas for a large population the fast fluctuations dominate (gray-shaded region). A Rescaled kurtosis
for the channel model (white diamonds) and the diffusion model (black circles). The gray symbols display simulations where the adaptation was replaced by an effective colored noise as before but with the additional white noise input. The case of an inverse Gaussian is indicated by the dotted line. B Corresponding serial correlation coefficient at lag one. The zero line is indicated by the dotted line. The adaptation time constant was chosen as
, other parameters as in Fig. 2.
Figure 11.
ISI histograms of the Traub-Miles model – deterministic vs. stochastic adaptation.
A The ISI densities of the Traub-Miles neuron model with a deterministic M-type adaptation current () and white noise driving (Eq. (115) – gray bars) is shown along with an inverse Gaussian (Eq. (64)) with the same mean and CV (dashed lines). To keep the firing rate at about
the external driving current was adjusted from top to bottom according to
,
,
,
(in
). Noise intensity
in units of
. B The ISI densities of the Traub-Miles model in the presence of a stochastic M-type adaptation current (Eq. (116) – gray bars) is shown along with an inverse Gaussian (Eq. (64)) with the same mean and CV (dashed line). Here, the external driving current was in all cases
.
Figure 12.
Comparison of the ISI statistics of the Traub-Miles model – deterministic vs. stochastic adaptation.
A Rescaled skewness (Eq. (61)) and B rescaled kurtosis
(Eq. (62)) as a function of the coefficient of variation (CV). For stochastic adaptation (Eq. (116),
– black circles) the number of channels was varied from
to
; for deterministic adaptation (Eq. (115),
– gray squares), the noise intensity was varied from
to
. The corresponding inverse Gaussian statistics (Eq. (64)) is indicated by the dotted line. C, D show the rescaled kurtosis and the serial correlation coefficient (Eq. (63)) at lag 1 as a function of the time scale separation
. Stochastic adaptation (
) and deterministic adaptation (
) are marked as in A,B. E,F The serial correlation coefficient
as a function of the lag
for different time constants
in
as indicated (E deterministic adaptation, F stochastic adaptation;
and
as in C,D). G The rescaled kurtosis
in the mixed case at a fixed amount of white noise (
) and varying channel numbers
. H The corresponding values of the serial correlation coefficient at lag one. The intersection of the
curve with the zero line (dotted line) defines the adaptation-noise dominated regime (white region) and the white-noise dominated regime (gray-shaded region). The units of the noise intensities are
. For stochastic adaptation
. For deterministic adaptation
was adjusted to result in a firing rate at around
. For
the current
was
. With increasing noise intensity
decreased to
for
.
Figure 13.
Channel model with voltage-dependent transition rates.
A In a simple model of a M-type adaptation channel, the slow voltage-gated adaptation current is mediated by a population of slow ion channels that can be either in an open or a closed state. In the activation state (), channels open with rate
, in the deactivation state (
) channels close with rate
. B Both transition rates show a sigmoidal dependence on the voltage (Eq. (24) and (27),
).
Figure 14.
Illustration of the deterministic dynamics of the adaptive PIF neuron.
The flow in the phase space spanned by possesses a stable limit cycle, which consists of the voltage drift from
at
to the threshold (A), the following increase of
by an amount
(B) and the subsequent voltage reset (C). The period of the limit cycle
is solely due to the time of the process A, whereas the processes B and C occur instantaneously. A trajectory starting off the limit cycle at
reaches the threshold at the time
. The deviation from
after one period has an absolute magnitude
.