Radial Gaussian

Creates connections using a Gaussian decay function that respects neuron polarity. The probability of making a connection drops off according to a Gaussian distribution centered on each neuron, scaled differently according to the polarity of the source and target neurons. The probability of forming a connection between a neuron \(a\) with polarity \(x\) and another neuron \(b\) with polarity \(y\) is given by:

\[P(a, b) = \min\left(C_{xy} \cdot \exp\left(-\left(\frac{D(a, b)}{\lambda}\right)^2\right), 1\right)\]

where:

  • \(D(a, b)\) is the Euclidean distance in pixels
  • \(C_{xy}\) is the scalar constant describing probabilities for making connections between neurons of polarity \(x\) and \(y\)
  • \(\lambda\) represents the standard deviation with respect to distance for the Gaussian drop-off

The larger any of the constants is relative to the others, the more likely a connection of that type will occur. For example, if \(C_{ee}\) is 0.2 and \(C_{ii}\) is 0.1, then excitatory-to-excitatory synapses are twice as likely as inhibitory-to-inhibitory synapses.

Parameters

  • Distance drop-off: Roughly the average connection distance in pixels. Lambda (\(\lambda\)) is a regulating constant governing overall connection density. Higher values create denser connections over larger distances.

  • Exc. → exc. constant: Controls the probability of forming a connection between two excitatory neurons.

  • Exc. → inh. constant: Controls the probability of forming a connection from an excitatory neuron to an inhibitory neuron.

  • Inh. → exc. constant: Controls the probability of forming a connection from an inhibitory neuron to an excitatory neuron.

  • Inh. → inh. constant: Controls the probability of forming a connection between two inhibitory neurons.

  • No polarity constant: Governs the probability of forming connections for neurons without explicit polarity.

Polarity Behavior

Radial Gaussian is unique among connection strategies in that it explicitly uses neuron polarity in its connection logic, not just for weight signs. The strategy checks both source and target neuron polarity and uses different connection constants (\(C_{ee}\), \(C_{ei}\), \(C_{ie}\), \(C_{ii}\)) for each combination.

This means:

  • The excitatory/inhibitory ratio setting is ignored (the polarity constants control the ratio)
  • Neurons should be pre-polarized for this strategy to work as intended
  • The four polarity constants let you control connection patterns (e.g., strong E→E connections, weak I→I connections)
  • Weight signs automatically follow source neuron polarity

For non-polar neurons, the No polarity constant is used.