I am currently advising seven Ph.D students and co-advising three Ph.D students with other faculty members in the department. All students are pursuing problems motivated by complex biological phenomena. They are:

• Ami Greenberg,
• Nathan VanderKraats,
• Nicholas Fisher,
• Venkatakrishnan Ramaswamy,
• Karthik S. Gurumoorthy,
• Subhajit Sengupta, and
• Ajit V. Rajwade with Prof. Anand Rangarajan.
• Santhosh Kodipaka with Prof. Baba Vemuri.
• Ritwik Kumar with Prof. Baba Vemuri.

• Hailong Meng graduated with a Ph.D in the summer of 2006.

Perspective

The abstract computational device that underlies a digital computer is a Finite State Automaton (or a Turing Machine if one assumes infinitely expandable memory). Our familiarity with the artifact, the digital computer in all its forms, is ultimately founded on this knowledge.

Interest in the nature of the human mind can be traced to the beginning of recorded history. There is now general consensus that the brain is implicated in cognition and behavior. If we are to decipher the nature of the human mind (and this includes all perception: visual, auditory, olfactory, tactile, or proprioceptive, higher order faculties such as language comprehension, and the most abstract of qualities such as attention), we must first consider the corresponding problem with regard to the brain: What abstract device underlies this physical system.

Research Objective

My principal research objective has been to determine the nature of the abstract computational device that recurrent systems of spiking neurons in the brain are a physical embodiment of. My approach has been to address the issue in stages, namely, to first determine the salient dynamical characteristics of such systems, and subsequently, to exploit this knowledge to arrive at a principled resolution of the matter.

I have been pursuing the following questions with regard to the dynamics of systems of spiking neurons in the brain.

• Are there coherent spatio-temporal structures in the dynamics of neuronal systems that can denote symbols.

  My usage of the term "symbol" conforms with the limited notion of a symbol as used in Computer Science---discrete states that mark a computational process regardless of representational content, if any, and not the notion of a symbol in the greater sense of the word---the physical embodiment of a semantic unit, as used in the Cognitive Sciences. The question therefore does not presuppose a position on the contentious issue of Representationalism.

• If such structures exist, what restrictions do the dynamics of the system at the physical level impose on the dynamics of the system at the corresponding abstracted symbolic level.

In order to address these questions, I have formulated an abstract dynamical system that models recurrent systems of spiking biological neurons (Banerjee, 2001a). The abstract system is based on a limited set of realistic assumptions and in consequence accommodates a wide range of neuronal models. I have evaluated the viability of the system by conducting extensive simulation experiments with the system set to model a typical column in the cortex. The characteristic behavior of the system is akin to that observed in neurophysiological experiments.

A thorough understanding of the dynamical behavior of the system can be achieved only by way of a comprehensive formal analysis of its dynamics. My approach has been to address the problem in stages. I began by analyzing the dynamics of the system under stationary conditions with either no input or input given by a stationary process. My efforts culminated in, among other results, a formal demonstration of the fact that under normal operational conditions, the dynamics of a typical neocortical column is governed by attractors that are not only almost surely (with probability 1) chaotic but are also potentially anisotropic (Banerjee, 2001b, 2003).

The general case must, however, address a problem of far greater complexity---the dynamics of systems with unconstrained inputs. This requires a mathematical framework that allows for the formal analysis of the transient dynamics of non-autonomous systems. A crucial question in this context is whether there is a formal counterpart to the concept of an attractor, an issue I am currently investigating.

Recent advances in neurophysiological research have begun to identify the manner in which synaptic efficacy changes as a function of the temporal relationship between the afferent spikes arriving at a synapse and the efferent spikes generated by the neuron. How these changes impact the noted set of attractors in the phase-space of the system, is however an issue that remains to be explored (Banerjee,2001c).

Within the framework of neuronal networks viewed as weakly perturbed systems, significant questions that remain unresolved include (i) what is the relation between the number of neurons in a system and the number of attractors that the corresponding phase-space can accommodate, (ii) what bearing, if any, does change in synaptic efficacy have on the number of attractors maintained by a system, and (iii) if the number of attractors does vary as a result of change in synaptic efficacy, what do formation, division, coalescence, and dissolution of attractors signify in terms of the symbol level dynamics of the system.