A Brief Guide to Neuronal Chaos

A considerable ammount of the world’s (scarce) scientific research funds has been allocated to the search for meaningfullchaotic patterns in many fields, from hard sciences (e.g., physicalsciences, engineering) to socioeconomic studies, with a wide range ofpromising and practical results. It appears that some of the tools ofthe science of nonlinear dynamics (and chaos) are also well-suited forstudies of biological phenomena, neuroscienceincluded. Indeed, such complex systems can give rise to collectivebehaviors which are not simply the sum of their components and involvehuge conglomerations of related units constantly interacting with theenvironment. There’s somewhat of a consensus that the activitiesundergone by neurons, neuronal assemblies and entire behavioralpatterns (e.g., after epileptic seizures), the linkage between them,and their evolution over time, cannot be understood in all itscomplexity and practical potential without these nonlinear techniques.

As an example, take the now classic Hindmarsh & Rose mathematical model of neuronal burstingusing 3 coupled first-order differential equations. Acomputer-simulated train of action potentials results in a pattern thatwould be interpreted as random on the basis of classical statisticalmethods, while a representation of interspike intervals reveals awell-ordered underlying generating mechanism (i.e., peak-to-peak dynamics).Rather naturally, the identification of nonlinear dynamics and chaos inan experimental neuronal setup is a very difficult task at variouslevels, far from the “clean” low dimensional chaos produced bycomputer/mathematical models. Firstly, there’s lack of stationarity onthe recorded signals, meaning that all the “parameters” of the(biological) system rarely remain with a constant mean and varianceduring measurements. This creates a not always viable need forprolonged and stable periods of observation. Secondly, collectedobservations generally exhibit a complex mixture offluctuations beyond the system itself, including those by theenvironment and those by the measurement equipment. For these purposesit’s helpful to start investigations by constructing a phase spacedescription 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 ofthe aforementioned technical difficulties neurobiologists have becomegradually more interested in practical issues such as the comparison ofdynamics of neuronal assemblies in various experimental conditions.With these less ambitious expectations, average (nonlinear)forecastings (e.g., of epileptic seizures) has been achieved in spiketrains (demonstrating determinism as a byproduct). Alternativelly, thesearch for Unstable Periodic Orbits(UPOs) in the reconstructed phase spaces has been fruitfull, which(paradoxically) results in an advantage if you desire to control aneuronal system to explore a large region of its phase space using onlya weak control signal. Recipe: Apply a (weak) control signal to forcethe system to follow closely any one of the identified UPOs, obtaininglarge changes in the long-term behavior with minimal effort – i.e., youcan select a given behavior from an “infinite” set and, if necessary,switch between them. Potential is unequivocal: Some abnormalities ofneuronal systems, ranging from differing periodicities to irregular”noise-like” phenomena could define a group of “dynamical deseases” of the brain.

Think of the 50×10^6+ epileptic people worldwide, ~20% of them notsufficiently helped by medications, taking the surgical removal of theseizure-focus as the last resort. Implants who (electrically) stimulatethe vagal nervehas also been used, but their action mechanism is uncertain, they haveseveral side effects, and they could potentially kindle new epilepticfoci in the area. Chaos control techniques might be used, with the advantage of requiring relatively infrequent stimulation of the tissue.

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