Neuronal Networks: A Discrete Model - Mathematical Concepts and Methods in Modern Biology - page 206

ganglia; however, in Parkinson patients, there is a significant increase in synchrony
among these neurons.
Another brain region where these types of firing patterns arise is the thalamus,
which plays an important role in sensory processing. Certain neurons within the
thalamus are thought to be involved in the transition between sleeping and waking
states [ 21 , 22 ]. Experiments have demonstrated that aswe fall deeper into sleep, certain
neurons within the thalamus become more synchronized with each other. Moreover,
computational models for sleep rhythms suggest that neurons within the thalamus
exhibit clustering during certain stages of sleep.
Finally, synchronous activity has been observed in cortical networks involved in
workingmemory [ 23 ]. It is tempting to view the brain (or some brain region) as a large-
scale dynamical system and a memory as an attractor of this dynamical system. If this
is the case, then it would be important to understand how the number of attractors, and
their basins of attraction, depend on network properties, including the intrinsic proper-
ties of agents (or neurons) in the system and network architecture. In particular, which
networks are capable of exhibiting a very large number of attractors (since one would
like to store a very large number of memories) and how do properties of attractors
depend on changes in parameters that may, for example, occur during learning?
We note that the neuronal systems described above share many properties, includ-
ing certain features of the neurons, synaptic connections, and network architecture.
Of course, there must be some important differences between these systems, because
they play such different functional roles in brain processing. However, from a mathe-
matical viewpoint, it may be possible to consider a general class of neuronal networks
that encompass each of these systems and develop an analytic framework that allows
us to characterize the firing patterns that emerge.
Our hope is that the analysis presented in this chapter is a first step toward develop-
ing such a framework. However, many challenges remain. Here, we characterized a
neuron by just two parameters: the firing threshold and the length of refractory period.
Obviously neurons are much more complicated biological objects with a myriad of
complex cellular processes. Whether or not a neuron fires an action potential may
depend on many factors. The firing threshold, for example, may depend on an inter-
nal “state” of the neuron, such as the concentration of some ionic species which may
change over time depending on how often the neuron fires. The strengths of synaptic
connections may also change over time, again depending on the firing rates of neu-
rons. It is not clear which of these details to include in a mathematical model, how
to incorporate these details into the theoretical framework described here, and how
difficult it would be to mathematically analyze discrete dynamical systems that do
include these details. Some progress in this direction is presented in [ 17 ].
It is important to emphasize that significant progress can be made only if there
is close dialog between mathematicians and experimental neuroscientists. It hardly
seems likely that a singlemathematical theory can be developed that takes into account
every possible detail that may go into a neuronal network model, including complex
properties of cells (such as their complex geometry) and all possible network architec-
tures. Collaborations with biologists help guide mathematicians to better understand