erfc_neuron – Binary stochastic neuron with complementary error function as activation function
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Description
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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 :math:`h` into the neuron is summed up,
passed through a gain function :math:`g` whose output is interpreted as
the probability of the neuron to be in the active (1) state.
The gain function used here is
.. math::
g(h) = \frac{1}{2} \mathrm{erfc} \frac{h - \theta}{\sqrt{2}\sigma}\;.
This corresponds to a McCulloch-Pitts neuron receiving additional
Gaussian noise with mean 0 and standard deviation :math:`\sigma`. The time
constant :math:`\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 :math:`\tau_m = dt \times N`, where :math:`dt` is the simulation
time step and :math:`N` the number of neurons in the original simulation with asynchronous
update. This ensures that a neuron is updated on average every :math:`\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 :math:`d`, the user has to specify
the delay :math:`d+h` upon connection, where :math:`h` is the simulation time step.
Parameters
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tau_m ms Membrane time constant (mean inter-update-interval)
theta mV threshold for sigmoidal activation function
sigma mV 1/sqrt(2pi) x inverse of maximal slope
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.. admonition:: Special requirements for binary neurons
As the ``erfc_neuron`` is a binary neuron, the user must
ensure that the following requirements are observed. NEST does not
enforce them. Breaching the requirements can lead to meaningless
results.
1. Binary neurons must only be connected to other binary neurons.
#. No more than one connection must be created between any pair of
binary neurons. When using probabilistic connection rules, specify
``'allow_autapses': False`` to avoid accidental creation of
multiple connections between a pair of neurons.
#. Binary neurons can be driven by current-injecting devices, but
*not* by spike generators.
#. Activity of binary neurons can only be recored using a ``spin_detector``
or ``correlospinmatrix_detector``.
References
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.. [1] Ginzburg I, Sompolinsky H (1994). Theory of correlations in stochastic
neural networks. PRE 50(4) p. 3171.
DOI: https://doi.org/10.1103/PhysRevE.50.3171
.. [2] McCulloch W, Pitts W (1943). A logical calculus of the ideas
immanent in nervous activity. Bulletin of Mathematical Biophysics,
5:115-133. DOI: https://doi.org/10.1007/BF02478259
.. [3] Morrison A, Diesmann M (2007). 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. DOI: https://doi.org/10.1007/978-3-540-73159-7_10
Receives
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CurrentEvent
See also
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:doc:`Neuron `, :doc:`Binary `