# erfc_neuron – Binary stochastic neuron with complementary error function as activation function¶

## Description¶

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 used here is

$g(h) = \frac{1}{2} \mathrm{erfc} \frac{h - \theta}{\sqrt{2}\sigma}\;.$

This corresponds to a McCulloch-Pitts neuron receiving additional Gaussian noise with mean 0 and standard deviation $$\sigma$$. 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 $$\tau_m = dt \times N$$, 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.

## Parameters¶

 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

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.

2. 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.

3. Binary neurons can be driven by current-injecting devices, but not by spike generators.

4. Activity of binary neurons can only be recored using a spin_detector or correlospinmatrix_detector.

## References¶

1(1,2)

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(1,2)

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