Table 1 Symbols with descriptions.
From: Constructing neural networks with pre-specified dynamics
Symbol | Description |
|---|---|
\(N_{neu}\) | Number of neurons in the network |
\(\textbf{z}\) | Vector containing the outputs (activations) of each neuron |
\(\textbf{u}\) | Vector of preactivations such that \(\mathcal{H}(\textbf{u})=\textbf{z}\) |
\(\textbf{W}_y\) | Matrix of synaptic weights between sensory inputs and neurons in the network. The ith row collects incoming weights from sensory inputs to the ith. neuron |
\(\textbf{W}_{r}\) | Matrix of synaptic weights between pairs of neurons in the network. The ith row collects incoming weights from all neurons to the ith neuron |
G | Labeled multidigraph with one node per network state and one arc for each transition. Arcs are labeled by the stimulus that triggers the transition |
\(N_{v}\) | Number of nodes in G |
\(N_{tran}\) | Number of transitions (arcs) in G |
v | A node in G |
V | The set of all nodes in G |
\(f_{s}\) | The function that maps arcs in G to their source nodes |
\(f_{t}\) | The function that maps arcs in G to their target nodes |
\(l_{G}\) | The function that maps arcs in G to their labels |
\(\textbf{G}\) | Matrix representation of graph G, with one row for transition. Columns contain stimulus, source node, and target node |
\(\textbf{Y}\) | Matrix of input vectors. The ith column collects the input vector \(\textbf{y}\) associated with the stimulus that triggers the ith transition in G (the ith row in \(\textbf{G}\)) |
Z | The set of network activation states \(\textbf{z}\). It maps to V in a one-to-one fashion |
\(\textbf{Z}_{s}\) | Matrix of activation states, when acting as sources in G. The ith column collects the activation states associated with the source node of the ith transition/row in \(\textbf{G}\) |
\(\textbf{Z}_{t}\) | Matrix of activation states, when acting as targets in G. The ith column collects the activation states associated with the target node of the ith transition/row in \(\textbf{G}\) |
\(\textbf{U}\) | Matrix of preactivations, such that \(\mathcal{H}(\textbf{U})=\textbf{Z}_{t}\) |
\(\varvec{\delta }_{i,j}\) | Equal to \(\textbf{W}_{y}(\textbf{y}_{s_{i}}-\textbf{y}_{s_{j}})\). Its length is \(N_{neu}\). It collects the difference between contributions of stimulus \(s_{i}\) and \(s_{j}\) to preactivations reached after any transition |
D | Labeled multidigraph. Its nodes are contained in V. Two nodes are connected if they are reached through stimuli \(s_{i}\) and \(s_{j}\) from a common source node in G. Arcs are labeled by \(\delta _{i,j}\) |
\(g_{s}\) | Function that maps arcs in D to their source nodes |
\(g_{t}\) | Function that maps arcs in D to their target nodes |
\(l_{D}\) | Function that maps arcs in D to their source labels |
C | The set of all labels/deltas in D |
P | A partition of C |
\(V_{trav}(C_{i})\) | A set that collects (source, target) node pairs of arcs in D, whos labels are in element \(C_{i}\) of partition P |
