Caputron

Implements Caputron, a Caputo fractional-order optimizer for Tempotron-like spiking neural networks.

Caputron replaces the first-order Tempotron gradient with a Caputo fractional derivative of order \(\alpha \in (0,1)\). The standard Tempotron update scales each weight by the kernel contribution \(K\) and the loss sign; Caputron multiplies this gradient by a power-law factor \((\theta - c)^{1-\alpha}\) with a Gamma-function normalization, where \(c\) is the per-neuron minimum weight that serves as the lower terminal of the fractional integral. As \(\alpha \to 1\) the factor reduces to the ordinary derivative, while smaller orders inject a memory-dependent, weight-magnitude-aware rescaling that the authors interpret as an adaptive normalization.

\[ \begin{aligned} c_i &= \min_j \theta_{ij}, \\ \theta_{t+1} &= \theta_t - \eta \, s_t \odot K \odot \frac{(\theta_t - c)^{1-\alpha}}{\Gamma(\alpha)\,(1-\alpha)}, \end{aligned} \]

where \(\theta\) are the synaptic weights, \(\eta\) the learning rate, \(\alpha \in (0,1)\) the Caputo derivative order, \(K\) the postsynaptic kernel contribution, \(s_t\) the Tempotron loss sign (\(+1\) for a missing required spike, \(-1\) for a spurious spike, \(0\) otherwise), \(c_i\) the minimum weight over the inputs of neuron \(i\), \(\Gamma\) the Gamma function, and \(\odot\) elementwise multiplication.

Reference: Natabara Máté Gyöngyössy, Gábor Erős, János Botzheim, "Exploring the Effects of Caputo Fractional Derivative in Spiking Neural Network Training", Electronics 2022. https://doi.org/10.3390/electronics11142114

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