Take a simple MLP. Let $x$ be the layer activation. $X^0$ is the input, $X^1$ is the second layer (first hidden layer). These are vectors, indexed like $x^a_i$ .
Then $X^1 = W X^0$ or $x^1_j = \phi(\Sigma_{i=1}^N w_{ij} x^0_i)$ . $\phi$ is the nonlinear activation function (ReLU, sigmoid, etc.)
In standard STDP the learning rule follows $\Delta w \propto f(x_{pre}(t), x_{post}(t))$ or if layer number is $a$$\Delta w^{a+1} \propto f(x^a(t), x^{a+1}(t))$
(but of course nobody thinks there 'numbers' on the 'layers' of the brain -- this is just referring to pre and post synaptic).
In an artificial neural network, $\Delta w^a \propto - \frac{\partial E}{\partial w_{ij}^a} \propto - \delta_{j}^a x_{i}$ (Intuitively: the weight change is proportional to the error propagated from higher layers times the input activity) where $\delta_{j}^a = (\Sigma_{k=1}^{N} w_{jk} \delta_k^{a+1}) \partial \phi$ where $\partial \phi$ is the derivative of the nonlinear activation function, evaluated at a given activation.
$f(i, j) \rightarrow [x, y, \theta, \phi]$
$k = 13.165$
$x = round(i / k)$
$y = round(j / k)$
$\theta = a (\frac{i}{k} - x) + b (\frac{i}{k} - x)^2$
$\phi = a (\frac{j}{k} - y) + b (\frac{j}{k} - y)^2$
PMID-27690349Nonlinear Hebbian Learning as a Unifying Principle in Receptive Field Formation
Here we show that the principle of nonlinear Hebbian learning is sufficient for receptive field development under rather general conditions.
The nonlinearity is defined by the neuronâ€™s f-I curve combined with the nonlinearity of the plasticity function. The outcome of such nonlinear learning is equivalent to projection pursuit [18, 19, 20], which focuses on features with non-trivial statistical structure, and therefore links receptive field development to optimality principles.
$\Delta w \propto x h(g(w^T x))$ where h is the hebbian plasticity term, and g is the neurons f-I curve (input-output relation), and x is the (sensory) input.
The relevant property of natural image statistics is that the distribution of features derived from typical localized oriented patterns has high kurtosis [5,6, 39]
Model is a generalized leaky integrate and fire neuron, with triplet STDP