We show some examples below of adaptivity in an N cell network of coupled odd logistic maps \(f(x) = a x (1  x^2)\). (We should emphasize that what we show is but a small sample of some of the remarkable dynamics that can occur!) The odd logistic map is scaled to give a selfmap of [0,1], computations are done mod 1 and the coefficient \(a \in [3,3]\). We consider cases where cells are identical and approximately identical. Coupling strengths (weights) are limited to lie in the range [0, 2.00] and then multiplied by c/number of cells where c > 0. Initial conditions are chosen randomly in the interval [0.1,0.8]; initial weights are chosen randomly and lie in [0.3,0.8]. Coupling is either alltoall or with a given sparseness which ranges zero (no coupling) to 1 (alltoall). When we rescale we keep \(c \times \text{sparseness}\) constant. We show dynamics of the cells and also weight averages for each node. The weight average measures the average over all inputs to a given node. It is also possible to show the evolution of the time averages of weights (we don't show that here). Currently we can simulate between \(N=4\) and \(N=10,000\) cells. We use various types of adaption (a total of 12 currently) including 'global' schemes based on work of ItoKaneko, 2002, 2003 and what we term 'fake STDP'. For the images we show here, we strengthen weights (both ways) between cells i and j if states of i and j are close and weaken if they are not close (we call an asymmetric directed version of this rule fake STDP). Typically we use adaptation of the form \[w^{n+1}_{ij} = (1 + C (1  Dx_i  x_j))w^n_{ij},\] where \(C,D > 0\) and \(x_i\) denotes state of cell i at time n (or n+1), and \(w^n_{ij}\) is the weight at time n for the connection from cell j to cell i. We note that the size of this computation per iteration grows like \(aN^2\), where a is independent of N (a is at least 10). This is a significant computation for a 'desktop' computer when N = 10,000 and we do say 12,000 iterations (order \(10^{13}\) computations). Almost all the computations we show here were done in extended precision (long double) and are sensitive to small quantities of the order \(10^{4900}\). This is necessary to pick up some of the heteroclinic phenomena over extended time periods. The software is based on prism, used in the past for computation of symmetric attractors and patterns, and is partly parallelized (threaded). The current version includes a tuning program that allows for processor dependent optimization of speed in multithreaded applications. Images shown here were all computed on a i7 990X custom built machine and typically used from 110 cores. In Figures 1 and 2 we assume 200 cells, 12184 iterations. We assume identical odd logistic maps with a = 2.31. Figure 1 shows dynamics, Figure 2 the evolution of averages of in weights from each cell.

We remark the complexity of the initial transient which shows characteristic features of a heteroclinic cycle (between chaotic sets). However, the heteroclinic phenomena is closely tied to the adaptivity. Run for a sufficiently long time, the network will reach a `steady state'. See Figure 3 below which shows dynamics over an interval of 121,840 iterations, starting at 365,520 iterations. Further iteration leaves the picture unchanged.
The transient and long term structure appears relatively independent of the number of cells. In the next two figures we show the first 150,000 iterations of a network with 500 cells. Parameters are similar to those of the 200 cell network.
In the next set of figures, which are all for a 1000 cell network, we (a) break the identical cell structure and allow for uniform distribution of the coefficients of the odd logistic map in a small interval, (b) show the effects of decreasing the sparseness of the coupling.
First we show the results when we assume identical odd logistic maps and all to coupling.
In Figure 10, we have identical cells but the sparseness of coupling is 0.9. As can be seen the transient behaviour is relatively unaffected.
In the next two figures, we show network adaptation using a different adaptive scheme (bangbang type control). We assume that weights evolve according to \[ w_{ij}^{n+1} = (1 + W(\theta_i,\theta_j))w_{ij}^n, \] where \(W(x,y) = a\), if \( xy < \alpha\), \(W(x,y) = b\) if \(xy \geq \alpha\), where \(0 < a < b < 1\) and typically \(a,b\) are small. We continue to assume that weights are constrained to lie in [0,2].