erfc_neuron - Binary stochastic neuron with complementary error  
function as activation function.
The erfc_neuron is an implementation of a binary neuron that
is irregularly updated at Poisson time points. At each update
point the total synaptic input h into the neuron is summed up,
passed through a gain function g whose output is interpreted as
the probability of the neuron to be in the active (1) state.

The gain function g used here is

g(h) = 0.5 * erfc (( h - theta_ ) / ( sqrt( 2. ) * sigma)).

This corresponds to a McCulloch-Pitts neuron receiving additional
Gaussian noise with mean 0 and standard deviation sigma.
The time constant tau_m is defined as the mean of the
inter-update-interval that is drawn from an exponential
distribution with this parameter. Using this neuron to reproduce
simulations with asynchronous update (similar to [1,2]), the time
constant needs to be chosen as tau_m = dt*N, where dt is the simulation time
step and N the number of neurons in the original simulation with
asynchronous update. This ensures that a neuron is updated on
average every tau_m ms. Since in the original papers [1,2] neurons
are coupled with zero delay, this implementation follows that
definition. It uses the update scheme described in [3] to
maintain causality: The incoming events in time step t_i are
taken into account at the beginning of the time step to calculate
the gain function and to decide upon a transition. In order to
obtain delayed coupling with delay d, the user has to specify the
delay d+h upon connection, where h is the simulation time step.

tau_m double - Membrane time constant (mean inter-update-interval) (ms)
theta double - threshold for sigmoidal activation function (mV)
sigma double - 1/sqrt(2pi) x inverse of maximal slope (mV)

SpikeEvent, PotentialRequest  
This neuron has a special use for spike events to convey the
binary state of the neuron to the target. The neuron model
only sends a spike if a transition of its state occurs. If the
state makes an up-transition it sends a spike with multiplicity 2,
if a down transition occurs, it sends a spike with multiplicity 1.
The decoding scheme relies on the feature that spikes with multiplicity
larger 1 are delivered consecutively, also in a parallel setting.
The creation of double connections between binary neurons will
destroy the decoding scheme, as this effectively duplicates
every event. Using random connection routines it is therefore
advisable to set the property 'multapses' to false.
The neuron accepts several sources of currents, e.g. from a

[1] Iris Ginzburg, Haim Sompolinsky. Theory of correlations in stochastic
neural networks (1994). PRE 50(4) p. 3171
[2] W. McCulloch und W. Pitts (1943). A logical calculus of the ideas
immanent in nervous activity. Bulletin of Mathematical Biophysics, 5:115-133.
[3] Abigail Morrison, Markus Diesmann. Maintaining Causality in Discrete Time
Neuronal Simulations. In: Lectures in Supercomputational Neuroscience,
p. 267. Peter beim Graben, Changsong Zhou, Marco Thiel, Juergen Kurths
(Eds.), Springer 2008.

May 2016  
Authors: Jakob Jordan, Tobias Kuehn
SeeAlso: Source: