Computational Neuroscience
Phase Resetting Curve (PRC) measures the advance/delay of subsequent spike/burst due to an external excitatory/inhibitory perturbation received by a neuron.
Background:
In a neural network, the most significant effect of a pre-synaptic stimulus consists in changing the firing rate of the post-synaptic neuron by either advancing or delaying the subsequent spikes (see Figure 1).
Figure 1
(A1) Transient change in the intrinsic period Pi of neural oscillators. In the case of Type I Morris-Lecar (ML) model neurons, a rectangular inhibitory input induces a significant hyper-polarization that delays the spike immediately following the perturbation and lengthens the time interval between the spikes to P1 (dashed trace). If the neuron is left alone, then the transient change in the spiking time dies out after a few cycles and the neuron returns to the unperturbed limit cycle with the original intrinsic period Pi
(A2) The first order PRC for Type I ML model neuron subject to a rectangular pulse of inhibitory current with amplitude - 0.10 (arbitrary units) and duration of 2.5 ms shows that the subsequent spike is delayed (positive resetting) due to incoming inhibition.
If the free running oscillation is stable, meaning that there is a stable limit cycle in the phase space, then it is useful to express the amount of advance or delay induced by a certain pre-synaptic input not in terms of stimulus time ts, which is the time interval between a spike of the driven neuron and the subsequent input received from the driving neuron, but in terms of the stimulus phase j = ts/Pi, where Pi is the intrinsic (free running) period of driven oscillator and j is called the stimulus phase. By convention, the transient phase resetting induced during current cycle, which is a function of the phase at which a neuron receives a perturbation from another neuron is defined by F1(j) = P1/Pi - 1. The definition of the first order phase resetting implies that a negative value of F(j) means that the transiently modified time interval between two successive spikes is shorter than the intrinsic period, and, therefore, the subsequent spike is advanced compared to its unperturbed position, whereas a positive value of the PRC refers to a delay of the subsequent spike compared to the intrinsic firing period, but different research groups use different sign conventions. The transient effect of synaptic perturbations on subsequent cycles is measured by higher (second, third, etc.) order phase resettings. For example, the second order phase resetting is defined by F2(j) = P2/Pi - 1.
Questions:
- how the PRC scales with the amplitude and duration of the pre-synaptic perturbation?
- what is the relationship between the bifurcation structure of the model and the PRC? Is a Type I spiking neuron always advancing the next spike? What about Type II excitability and the corresponding PRC?
- what is the effect of additive noise on PRC?
Central Pattern Generator (CPG)
are autonomous units capable of generating rhythmic and adaptive output without higher level inputs.
Background
The CPGs are small biological circuit (pacemakers) responsible for autonomous activities such as flying, swimming, walking, respiration and many other activities that require fast adaptation to environmental changes without cerebral cortex intervention.
Questions:
I am interested in developing existence and stability criteria CPGs. We previously tested the effectiveness of the theoretical method and successfully predicted the 1:1 pattern generated in hybrid circuits formed with PD neuron from stomatogastric ganglion of lobster Homarus Americanus coupled with a model neuron through dynamics clamp (Biophysical Journal, 2004).
For the case of small amplitude perturbations of the neural oscillators, which corresponds to weakly coupled network, we found analytic solution for the phase resetting induced by arbitrary stimuli.
The analytical method is based on a topological conversion between the phase space displacement and the temporal phase shift, and gave very good agreement with the computational results for type I excitability (published in Neural Computation).
We subsequently extended the applicability of the topological methods using phase space reconstruction based on Taken's method to include strong coupling in neural networks.
The new generalization of the phase resetting method was in very good agreement with both the computer simulations and the experimental results (published in Biophysical Journal and Neurocomputing).
Besides single-neuron response to external inputs, I was also interested in using the phase resetting method to predict the existence and the stability of steady firing patterns in small networks (CPG).
The stability criterion for networks included second order corrections for the firing phase under external perturbations and was successfully tested in computational models (published in the International Journal of Differential Equations) and later on in experiments performed on the pyloric circuit of lobster (published in the Biophysical Journal).
I intend to further develop analytical criterion for the existence and the stability of the phase-locked modes in networks of neural oscillators. A possible way of generalization of the existing stability criteria is through periodic constraints on coupled oscillators network and the use of circulant matrices theory. Although the periodicity constraint is physiologically justifiable for some populations of neurons a hierarchical structure with vertical couplings is possible only under Toepliz coupled maps theory. In this respect, it seems much easier to obtain a stability criterion for synchronous state in large populations of neurons and subsequently refine it to other phase-locked modes.
Another significant direction of research I intend to pursue is the VLSI implementation of the CPG for robotic application. During the last decade the neuromorphic engineering obtained significant successes in design of small legged robots inspired from biology with a very simple but adaptive motion controller using hexapods CPG blueprint. I intend to develop a new class of motion controller for legged robots based on phase resetting correction and not the traditional silicon neurons implementation. The advantage is a much faster response to environmental changes and wider control parameter range. The price paid for such an enhancement in the overall performance of the artificial CPG is the reduced number of tasks (degrees of freedom) available for the envisioned robot. At this time I imagine that a simple and probably not very efficient way of solving the complexity of the tasks drawback is the usage of finite automata implementation in simple logical circuits either statically written in traditional ROM or dynamically assembled in nano-circuits. This research project is closely related to my interest in artificial intelligence and functional self-organization of mobile agents. The neuromorphic CPG is only the first and the simplest layer of robot-environment interaction. The second layer I envision is based on memory registers and decision making algorithms I already developed and tested for software mobile agents (robot-like-ants).
I am also involved in developing a new computer model for dopamine neurons, which are primarily elements in memory and learning models, but are also involved in reward-based activities and drug addiction.
Complex Systems
I am particularly interested in analytical solutions of coupled nonlinear evolution equations for complex systems and the corresponding cellular automata implementations. The focus of the traditional fields of Control Theory and Artificial Intelligence recently shifted from hierarchical, centralized, and supervised controllers toward autonomous, decentralized and unsupervised decision making and coordination, the so called Swarm Intelligence. A swarm is a set of (mobile) agents which are liable to communicate directly or indirectly with each other by acting on their local environment, and which collectively carry out a distributed problem solving. Decentralized solutions based on swarm intelligence were proposed and tested on small scale networks in telecommunication. The artificial "ants", tiny programs that roam around networks behaving much like their living counterparts, updating the routing tables of the telecommunication networks and instantly redirecting the packages across the network in order to avoid traffic jams.
To implement software mobile agents I used a cellular automata approach based on a stochastic learning algorithms, and I proved that there is an emergent behavior at the level of the team by measuring the increase in the environment's degree of organization (published in Journal of Physics A: Mathematical and General). As a concrete application, I modeled the cooperative behavior of the immune system response to carcinoma using self-organizing cellular automata paradigm (published in Bioinformatics).
I would like to further develop a global model of carcinoma based on both physiological measurements and the cellular automata implementations.
Non-equilibrium and Non-extensive Statistics
Recent progresses in the complex systems theory reopened a febrile search for microscopic description of self-organization in the framework of the statistical mechanics. The usual additivity requirement for the thermodynamic quantities obtained by averaging method puts constraint on symmetries of the phase space of the system under consideration. Experimental measurements in colossal magneto-resistance manganites, amorphous and glassy nano-clusters and high-energy collision processes revealed that such systems are able to remain in non-equilibrium states for very long periods compared to the microscopic time scale of the underlining processes. The most disputed feature of such processes is the non-additivity of thermodynamic quantities, such as entropy. It is well-known that the Boltzmann-Gibbs-Shannon entropy leads to entropy additivity rule, which means that the entropy of a system composed by independent parts is the sum of the entropies of its parts. In order to explain metastability and non-additivity of thermodynamic quantities alternative entropy definitions were proposed during the last decade. Tsallis statistics is one of the viable approaches since the entropy definition is both non-additive and concave. A fundamental distinction between the extensive Boltzmann-Gibbs-Shannon and Tsallis non-extensive statistics is the presence of the entropy index q, which is a measure of the distance from a steady equilibrium. In particular, all systems at equilibrium are described by Boltzmann-Gibbs-Shannon statistics, which coincides with Tsallis statistics description for unity universality coefficient.
I developed a generalized approach on entropy index computation based on generalized Hurwitz zeta functions. The analytical method developed allowed me to explicitly compute the entropy index for a quantum gas with liner distribution of energy levels. This application of the non-extensive statistics opened the possibility of explicitly connecting the microscopic dynamics and the macroscopic evolution of the system under consideration. From a dynamical point of view, the same results could be obtained by using the Kolmogorov approach in dynamics. Along this line, the multidimensional Lyapunov exponent could be computed along the phase space trajectory and fitted to a generalized exponential according to generalized Tsallis statistics. This is a straightforward way to obtain the multi-fractal spectrum of the entropy index. My intention is to establish a formal relationship between the dynamical approach of Lyapunov and the topological entropy formalism of Kolmogorov-Sinai.
Chaos and Nonlinear Dynamics
The history of nonlinear physics goes back to Kepler's works on planets motion and to the early days of calculus. The singularity theory, sometimes called catastrophe of bifurcation theory, was developing both through the interest of mathematicians, physicists and chemical engineers.
The main difficulty when dealing with any nonlinear system is that one rarely has the luxury of knowing the underlying equations of evolution.
There are few well-established techniques to help us gain some geometric intuition on the phase space topology mainly due to Taken's embedding theorem.
I successfully used the embedding theorem for phase space reconstruction of pyloric dilator membrane potential record (published in Biophysical Journal). I intend to further develop topological methods for periodic orbits embedding and implement efficient numerical algorithms for such cases. The problem of reconstructing the (quasi)periodic noisy trajectories is particularly challenging due to the strong sensitivity of the nearby trajectories to small fluctuations. In the same time, running a computer simulation of a chaotic system leads to continual accumulation of small errors on the order of machine roundoff, making the computed trajectory distinct from the true trajectory that is the goal of the simulation. Although the nonlinear phenomena are ubiquitous, the computational paradigms are still relying on the unrealistic hypothesis of numerical stability to small roundoff errors. One solution I envision for this problem is the implementation of quantum computation algorithms based on recently developed trends in parallel computing.
Besides computational solutions to nonlinear evolution equations, which is the main trend nowadays, I also developed analytical solutions based on Krilov's averaging techniques and time scale separations (published in Physical Review).
I would like to continue my analytical approach along the following particularly fruitful directions:
- The generalization of the averaging techniques based on curvilinear coordinates, which smoothly transforms the local bifurcation structure into a Riemann surface of zero curvature,
- The use of Hidden Markov Models to extract the essential information about the original phase space embedding, which proved particularly efficient in detecting the embedding dimension of the underlying dynamics,
- Improved standard least-square fits to estimate the structural parameter of a low-dimensional deterministically chaotic system (published in Chaos), and
- Implementation of proportional feedback control mechanisms for chaotic systems (published in Journal of Technical Physics).
Practical implementation of control meets serious difficulties because a system at the edge of chaos is sensitive even to small parameter variations. Action of global feedbacks on chaotic extended systems has been experimentally and theoretically investigated for lasers, gas discharges, semiconductors, populations of electrochemical oscillators, and surface chemical reactions.
Theoretical Condensed Matter
In their 1987 work, Bak, Tang and Wiesenfeld showed that certain open, dissipative, spatially extended systems spontaneously achieve a critical state characterized by power-law distribution of event sizes. They called this phenomenon Self-Organized Criticality (SOC) and illustrated it using a simple two-dimensional cellular automaton model for sand-piles. Since then, many natural phenomena that have been connect to SOC, including but not limited to earthquakes, evolution, interface dynamics, vortices in superconductors etc. Non-equilibrium surface growth and interface dynamics represent an effervescent area of research with a large number of discrete atomistic growth models and stochastic growth equations exhibit generic scale invariance characterized by power law behavior (SOC). Diffusion Limited Aggregation (DLA) is the main paradigm used to solve the aggregation of clusters via diffusion and attachment. Electrochemical deposition and Molecular Beam Epitaxy are two examples of general models developed in the context of DLA and implemented using cellular automata algorithms. Molecular Beam Epitaxy is applied, for example, in the growth of layered semiconductor heterostructures for electronic devices or in the development of thin magnetic films for novel storage media, plays also a significant role as a tool in the design of nano-structures, such as Quantum Wires or Dots.
The computation model I developed for DLA modeling incorporates the traditional Metropolis updating rule subject to detailed balance. The novelty of the proposed approach consist in connecting the macroscopic measures, such as surface roughness, to dynamic quantities, such us the exponent of the two-particle interaction potential. Our preliminary results indicate a deterministic relationship between the capacity fractal dimension and the exponent of the two-particle potential. Therefore, our method could be used in conjunction with the electron microscopy to determine the Hamiltonian at the microscopic level. I intend to extend the applicability of the method for the purpose of microscopic model prediction by including higher order fractal measures. Of special interest is also the development of numerically efficient implementation of the DLA cellular automata. The initial implementation used Pascal (because of the integrated graphic interface that allows instant graphic visualization of numerical results). More recently the whole implementation was ported to C language (for fast computation) and interfaced with Mathematica (for easy graphical visualization).
Use of Technology in the Classroom
I am a firm believer in the need for efficient teaching. In the recent decade, the advances in technology and the wide spread of computers have introduces new means for teaching. However, most of these techniques are used inefficiently and usually do not result in improved class outcome. My research interests here are related to the study of linking class presentation and assessment tools to classic educational methods. My research in this field materialized in three books designed to help students enhance their conceptual understanding of physics principles and strengthen their problem solving skills. I am currently developing a set of conceptual problems to be used in conjunction with the peer instruction method.