asymptoticdesign

Nonlinear dynamics and mathematical modelling of neuronal populations

The study of a dynamical system and its bifurcations can be used to investigate neuronal behaviour. The pupil light reflex (see summary description) has an important neuronal component and it can be observed non-invasively in well documented and well controlled conditions. There is a vast amount of published work on the pupil light reflex, its anatomy and importance in clinical neuroscience, and on the mathematical tools, like Delay Differential Equations, used to model it. The exact characteristics of the neuronal component of the pupil light reflex are still not fully known and mathematical modelling of the nonlinearities involved can bring a better understanding of the data obtained from measurements of the pupil response to light.
A biologically plausible model of the pupil light reflex with Delay Differential Equations, extending a model by A. Longtin, John Milton et al. from McGill University, Canada, is introduced and analysed in Nonlinear shunting model of the pupil light reflex, by P. C. Bressloff, C. V. Wood and P. A. Howarth, Proc. Roy. Soc. B vol. 263, 1996 and Spontaneous oscillations in a nonlinear delayed-feedback shunting model of the pupil light reflex (in PDF format), by P. C. Bressloff and C.V.Wood, Phys. Rev. E, vol. 58, 1998. A number of neuronal units are modelled by delayed differential equations as leaky integrators with shunting, meaning that the rate of change in the membrane potential depends on the difference between the current membrane potential and the equilibrium value. This way the mathematical model implements the logarithmic-like saturation of a neuron's response to very large input. The time delay in the equations is the delay between the moment when the light stimulus reaches the retina and the neuron starts firing in response to this stimulus. The scope of the model is to obtain maximum information about the neuronal population from measurements of the pupil size in response to light stimuli.

There is a subtle relation between the pupil area that is determined the iris muscle activity, and the retinal area responding to light, that determines the light flux. In normal conditions, the light stimulus has a spatial field larger than the eye and the responding area of the retina is determined by the pupil size (closed loop). There are also Maxwellian view experimental conditions, in which the light stimulus is a spot narrower than the smallest size that the pupil can reach, and the responding retinal area does not depend on the magnitude of the pupil area (open loop).

The parameters form a multi-dimensional continuous space. As they vary, the stability of solutions can change, a process studied by bifurcation theory for dynamical systems. An aspect of interest is the onset of oscillations, both slowly oscillating solutions (passing through the equilibrium point at time intervals larger than the maximum time delay), and rapidly oscillating solutions. Slowly oscillating solutions can be used to describe the pupil cycling at pupil-edge light stimulation and rapidly oscillating solutions can be used to investigate the pupillary hippus (fluctuations in pupil size at steady illumination).
The behaviour of the model is compared to data from measurements of the pupil light response for a better understanding of the parameters and nonlinearities involved and their importance in medical diagnosis. There is a large amount of noise and variation in the experimental data and numerical simulations are not sufficient in understanding the qualitative behaviour of the model in a multi-dimensional parameter space.

The model considers at this stage only the parasympathetic branch of the pupil light reflex, concentrating on the neural component. The equations do not take into account interactions between neurons, that are not very well known. The model is like a jigsaw-puzzle: as more biologically plausible features are added (nonlinearities, more neuronal units, interactions between neurons, interactions between the paths corresponding to each eye, the inhibitory activity related to the sympathetic branch, a more detailed modelling of the retina and of the iris muscle, using Bayes probability theory to estimate parameters), the model fits better the experimental data and its parameters and nonlinearities can be better mapped to the real components of the reflex loop, widely studied in medicine and ergonomics.

Acknowledgment - the title image is a 3-d orbit plot from Maple Picture Gallery, North Carolina State University.