A considerable ammount of the world's (scarce) scientific research funds has been allocated to the search for meaningfull chaotic patterns in many fields, from hard sciences (e.g., physical sciences, engineering) to socioeconomic studies, with a wide range of promising and practical results. It appears that some of the tools of the science of nonlinear dynamics (and chaos) are also well-suited for studies of biological phenomena, neuroscience included. Indeed, such complex systems can give rise to collective behaviors which are not simply the sum of their components and involve huge conglomerations of related units constantly interacting with the environment. There's somewhat of a consensus that the activities undergone by neurons, neuronal assemblies and entire behavioral patterns (e.g., after epileptic seizures), the linkage between them, and their evolution over time, cannot be understood in all its complexity and practical potential without these nonlinear techniques.

As an example, take the now classic Hindmarsh & Rose mathematical model of neuronal bursting using 3 coupled first-order differential equations. A computer-simulated train of action potentials results in a pattern that would be interpreted as random on the basis of classical statistical methods, while a representation of interspike intervals reveals a well-ordered underlying generating mechanism (i.e., peak-to-peak dynamics). Rather naturally, the identification of nonlinear dynamics and chaos in an experimental neuronal setup is a very difficult task at various levels, far from the "clean" low dimensional chaos produced by computer/mathematical models. Firstly, there's lack of stationarity on the recorded signals, meaning that all the "parameters" of the (biological) system rarely remain with a constant mean and variance during measurements. This creates a not always viable need for prolonged and stable periods of observation. Secondly, collected observations generally exhibit a complex mixture of fluctuations beyond the system itself, including those by the environment and those by the measurement equipment. For these purposes it's helpful to start investigations by constructing a phase space description of the underlying phenomenon (i.e., phase space reconstruction and embedding of a time series), usually plotting the relationship between successive events or time intervals (i.e., a Poincaré map), as most of the relevant signals are discrete ones.

And so what? Is that just public/media curiosity? In the light of the aforementioned technical difficulties neurobiologists have become gradually more interested in practical issues such as the comparison of dynamics of neuronal assemblies in various experimental conditions. With these less ambitious expectations, average (nonlinear) forecastings (e.g., of epileptic seizures) has been achieved in spike trains (demonstrating determinism as a byproduct). Alternativelly, the search for Unstable Periodic Orbits (UPOs) in the reconstructed phase spaces has been fruitful, which (paradoxically) results in an advantage if you desire to control a neuronal system to explore a large region of its phase space using only a weak control signal. Recipe: Apply a (weak) control signal to force the system to follow closely any one of the identified UPOs, obtaining large changes in the long-term behavior with minimal effort - i.e., you can select a given behavior from an "infinite" set and, if necessary, switch between them. Potential is unequivocal: Some abnormalities of neuronal systems, ranging from differing periodicities to irregular "noise-like" phenomena could define a group of "dynamical deseases" of the brain.

Think of the 50x10^6+ epileptic people worldwide, ~20% of them not sufficiently helped by medications, taking the surgical removal of the seizure-focus as the last resort. Implants who (electrically) stimulate the vagal nerve has also been used, but their action mechanism is uncertain, they have several side effects, and they could potentially kindle new epileptic foci in the area. Chaos control techniques might be used, with the advantage of requiring relatively infrequent stimulation of the tissue.

In a mathematical space whose coordinates represent the state of a dynamical system (i.e., a state space), periodic orbits are the set of equilibrium states. If all of the periodic orbits in this abstract dynamical landscape are unstable, the system's temporal evolution will never settle down to any one of them. Instead, system's behavior wanders incessantly in a sequence of close approaches to these orbits. The more unstable an orbit, the less time the system spends near it. Unstable Periodic Orbits (UPOs) form the "skeleton" of nonlinear dynamics, and one can build a model of a system by counting and characterizing its UPOs in a hierarchy of orbits with increasing periodicity. Model's accuracy can be improved by progressivelly adding longer-period orbits to the hierarchy. The dynamical landscape can then be tesselated into regions of the state space centered around these UPOs. Orbit locations and stability can also offer short-term predictions for the system's future states. This type of predictive model can be used for parametric control of nonlinear systems, whether they are chaotic or not. However, rigorous identification of UPOs from noisy experimental data is a difficult task.

There's a straightforward method for identification of UPOs which relies on the recurrence of patterns in state space, though that's a very rare event (i.e., a state repeatedly returning near an orbit) in the reality of short and nonstationary datasets. There's another method based on a local dynamics data transformation, which acts as a dynamical lens so that the new datasets are concentrated about distinct UPOs, helping to offset the usual scarcity of trajectories near UPOs. With the additional ability to identify complex higher period orbits by using only fragments of trajectories near those orbits, identification of UPOs was successfully achieved in various experimental settings, including epileptiform activities from the human cortex.

Tracking "parameter" changes from the inherently nonstationary data of, e.g., neurological systems, with UPOs has also been accomplished, as this is a strong requirement for UPO-based control of nonlinear systems. Furthermore, this tracking could be used to detect changes in system state due to intrinsic "parameter" variations (e.g., the transition to epileptic seizures), extrinsic effects (e.g., due to electromagnetic fields), or even perceptual discrimination.

The now classical Hindmarsh & Rose mathematical model of neuronal bursting was first published in peer-reviewed form back in 1984, by a scientific periodical named "Proceedings of The Royal Society of London. Series B, Biological Sciences". (This seminal paper is now only available online via a JSTOR's (probably institutional) subscription, so go check out with your university. Alternatively, go to your university's medical school library periodicals section.)

Hindmarsh & Rose work were initiated by the discovery of a neuronal cell in the brain of the pond snail Lymnae, which was initially "silent" (the molluscan burst neuron had been previously hyperpolarized to stop the bursting), but when depolarized by a short current pulse generated a burst that greatly outlasted the stimulus - i.e., an action potential followed by a slow depolarizing after-potential.

In seeking an explanation for the phenomena also observed in crustaceans and vertebrates, the research collaborators devised a system of 3 coupled (3-variable) first-order differential equations of the following form:

dx/dt = y - f(x) - z - I,
dy/dt = g(x) - y,
dz/dt = r(h1(x) - z),

where

(the first 3 variables listed below are coordinates which represent the states of the dynamical system - a single neuron in this case - varying over time)

x: (neuron) membrane potential,
y: potential of the ionic channels subserving accomodation,
z: the slow adaptation current which moves the voltage in and out of the inherent bistable regime and which terminates spike discharges,
r: the time scale of the slow adaptation current,
I: the applied current,
h1(x): the scale of the influence of the slow dynamics on membrane potential, which determines whether the neuron fires in a tonic or in a burst mode (when exposed to a sustained current input),
f(x): a cubic function,
g(x): not a linear function.

Depending on the values of the above parameters, neurons can be in a steady-state, they can generate a periodic low-frequency repetitive firing, chaotic bursts, or high frequency discharges of action potentials. Despite its inherent single cell description, Hindmarsh & Rose neurons can be linked by introducing equations accounting for electrical and/or chemical junctions which underlie syncronization in experiment material on the cooperative behavior of neurons that arises when cells belonging to large assemblies are coupled with each other. Depending on the degree of coupling between the neurons, such a linkage can lead to out/in phase bursting or to a chaotic behavior.