Abstract:
In recent years our understanding of cognition has become increasingly reliant on the tools of dynamical systems theory. Heteroclinic networks are one of these tools, and they can be found in a
number of different models of cognitive dynamics. One particular aspect of heteroclinic networks
which has not been explored to its full potential is the appearance of non-Markovian memory in
the presence of noise. This is of particular interest when modelling cognitive
processes, since there are frequently interdependencies between different cognitive functions.
This research will illuminate the use of memory in heteroclinic networks in two ways. Firstly, the
effect of noise on the dynamics near saddle equilibria of a heteroclinic network is investigated.
This is done by constructing a local map near a noisy saddle, and mapping different initial
distributions onto exit distributions and analytically deriving some of the statistical moments.
This local analysis yields new results concerning the global effect of memory in the network, in
particular, when the initial distribution depends on earlier interactions between the noise and the
network. We use the local map to determine conditions under which the dynamics associated with
the heteroclinic network are non-Markovian. In particular, we develop an algorithm to
determine a lower bound for the order of the associated Markov chain and calculate some transition
probabilities.
Secondly, a model of task-switching is built using a mixed heteroclinic and excitable network with
non-autonomous input. Task-switching is a good example of a cognitive process that
depends on memory since the performance of one task depends on the task that came before. A return
map is constructed to investigate the effect of input on the dynamics and numerical simulations of
the model are run. This numerical and theoretical analysis shows that non- autonomous
input can be used to produce a similar memory effect to noise, so that the time it takes to
complete a cycle of the network depends on whichever cycle was most recently completed.
Thus the model demonstrates that heteroclinic and excitable networks with input
can be used to reproduce the qualitative dynamics associated with task-switching.