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pp_psc_delta – Point process neuron with leaky integration of delta-shaped PSCs

Description

pp_psc_delta is an implementation of a leaky integrator, where the potential jumps on each spike arrival. It produces spike stochastically, and supports spike-frequency adaptation, and other optional features.

Spikes are generated randomly according to the current value of the transfer function which operates on the membrane potential. Spike generation is followed by an optional dead time. Setting with_reset to true will reset the membrane potential after each spike.

The transfer function can be chosen to be linear, exponential or a sum of both by adjusting three parameters:

\[rate = Rect[ c_1 * V' + c_2 * \exp(c_3 * V') ],\]

where the effective potential \(V' = V_m - E_{sfa}\) and \(E_{sfa}\) is called the adaptive threshold. Here Rect means rectifier: \(Rect(x) = {x \text{ if } x>=0, 0 \text{ else}}\) (this is necessary because negative rates are not possible).

By setting c_3 = 0, c_2 can be used as an offset spike rate for an otherwise linear rate model.

The dead time enables to include refractoriness. If dead time is 0, the number of spikes in one time step might exceed one and is drawn from the Poisson distribution accordingly. Otherwise, the probability for a spike is given by \(1 - \exp(-rate*h)\), where h is the simulation time step. If dead_time is smaller than the simulation resolution (time step), it is internally set to the resolution.

Note that, even if non-refractory neurons are to be modeled, a small value of dead_time, like dead_time=1e-8, might be the value of choice since it uses faster uniform random numbers than dead_time=0, which draws Poisson numbers. Only for very large spike rates (> 1 spike/time_step) this will cause errors.

The model can optionally include an adaptive firing threshold. If the neuron spikes, the threshold increases and the membrane potential will take longer to reach it. Here this is implemented by subtracting the value of the adaptive threshold E_sfa from the membrane potential V_m before passing the potential to the transfer function, see also above. E_sfa jumps by q_sfa when the neuron fires a spike, and decays exponentially with the time constant tau_sfa after (see 2 or 3). Thus, the E_sfa corresponds to the convolution of the neuron’s spike train with an exponential kernel. This adaptation kernel may also be chosen as the sum of n exponential kernels. To use this feature, q_sfa and tau_sfa have to be given as a list of n values each.

The firing of pp_psc_delta is usually not a renewal process. For example, its firing may depend on its past spikes if it has non-zero adaptation terms (q_sfa). But if so, it will depend on all its previous spikes, not just the last one – so it is not a renewal process model. However, if “with_reset” is True, and all adaptation terms (q_sfa) are 0, then it will reset (“forget”) its membrane potential each time a spike is emitted, which makes it a renewal process model (where “rate” above is its hazard function, also known as conditional intensity).

pp_psc_delta may also be called a spike-response model with escape-noise 6 (for vanishing, non-random dead_time). If c_1>0 and c_2==0, the rate is a convolution of the inputs with exponential filters – which is a model known as a Hawkes point process (see 4). If instead c_1==0, then pp_psc_delta is a point process generalized linear model (with the canonical link function, and exponential input filters) (see [5,6]_).

This model has been adapted from iaf_psc_delta. The default parameters are set to the mean values given in 2, which have been matched to spike-train recordings. Due to the many features of pp_psc_delta and its versatility, parameters should be set carefully and consciously.

Parameters

The following parameters can be set in the status dictionary.

V_m

mV

Membrane potential

C_m

pF

Capacitance of the membrane

tau_m

ms

Membrane time constant

q_sfa

mV

Adaptive threshold jump

tau_sfa

ms

Adaptive threshold time constant

dead_time

ms

Duration of the dead time

dead_time_random

boolean

Should a random dead time be drawn after each spike?

dead_time_shape

integer

Shape parameter of dead time gamma distribution

t_ref_remaining

ms

Remaining dead time at simulation start

with_reset

boolean

Should the membrane potential be reset after a spike?

I_e

pA

Constant input current

c_1

Hz/mV

Slope of linear part of transfer function in Hz/mV

c_2

Hz

Prefactor of exponential part of transfer function

c_3

1/mV

Coefficient of exponential non-linearity of transfer function

References

1

Cardanobile S, Rotter S (2010). Multiplicatively interacting point processes and applications to neural modeling. Journal of Computational Neuroscience 28(2):267-284 DOI: https://doi.org/10.1007/s10827-009-0204-0

2(1,2)

Jolivet R, Rauch A, Luescher H-R, Gerstner W. (2006). Predicting spike timing of neocortical pyramidal neurons by simple threshold models. Journal of Computational Neuroscience 21:35-49. DOI: https://doi.org/10.1007/s10827-006-7074-5

3

Pozzorini C, Naud R, Mensi S, Gerstner W (2013). Temporal whitening by power-law adaptation in neocortical neurons. Nature Neuroscience 16:942-948. (Uses a similar model of multi-timescale adaptation) DOI: https://doi.org/10.1038/nn.3431

4

Grytskyy D, Tetzlaff T, Diesmann M, Helias M (2013). A unified view on weakly correlated recurrent networks. Frontiers in Computational Neuroscience, 7:131. DOI: https://doi.org/10.3389/fncom.2013.00131

5

Deger M, Schwalger T, Naud R, Gerstner W (2014). Fluctuations and information filtering in coupled populations of spiking neurons with adaptation. Physical Review E 90:6, 062704. DOI: https://doi.org/10.1103/PhysRevE.90.062704

6

Gerstner W, Kistler WM, Naud R, Paninski L (2014). Neuronal Dynamics: From single neurons to networks and models of cognition. Cambridge University Press

Sends

SpikeEvent

Receives

SpikeEvent, CurrentEvent, DataLoggingRequest