Research Statement Statistical Physics Methods and Large Scale

Research Statement Maxim Shkarayev

Statistical physics methods and large scale computations In recent years, the methods of statistical physics have found applications in many seemingly un- related areas, such as in information technology and bio-physics. The core of my recent research is largely based on developing and utilizing these methods in the novel areas of applied mathematics: evaluation of error statistics in communication systems and dynamics of neuronal networks. During the coarse of my work I have shown that the distribution of the error rates in the fiber optical communication systems has broad tails; I have also shown that the architectural connectivity of the neuronal networks implies the functional connectivity of the afore mentioned networks. In my work I have successfully developed and applied methods based on optimal fluctuation theory, mean-field theory, asymptotic analysis. These methods can be used in investigations of related areas.

Optical Communication Systems Error Rate Statistics in Fiber Optical Communication Links. During my studies in the Ap- plied Mathematics Program at the University of Arizona, my research was focused on the analytical, numerical and experimental study of the statistics of rare events. In many cases the event that has the greatest impact is of extremely small likelihood. For example, large magnitude earthquakes are not at all frequent, yet their impact is so dramatic that understanding the likelihood of their occurrence is of great practical value. Studying the statistical properties of rare events is nontrivial because these events are infrequent and there is rarely sufficient time to observe the event enough times to assess any information about its statistics. This makes field experiments prohibitively expensive. Computer simulations of such events are impractical for the same reason: in order for Monte Carlo simulation to work one needs to develop a method of importance sampling. I have studied rare event phenomena in optical fiber communication systems. The quality of a communication system is described by a number that represents the probability of making an erroneous interpretation of the transmitted information. I studied the occurrence of errors in a system due to the presence of structural disorder of the fiber and temporal amplifier noise. Errors must be extremely rare in order for such a system to be useful. Moreover, structural structural disorder of the fiber changes slowly over time, changing the probability of receiving an error. Using the optimal fluctuation method from statistical physics, my collaborators and I defined the error statistics and showed that the distribution of error rates has broad tails; more specifically, the distribution was shown to be log-normal. Furthermore, previously unknown interplay of structural disorder and temporal noise was discovered and described. This result showed that describing the performance of the communication system by a mean probability of error occurrence is incomplete, and the full distribution must be taken into account in order to avoid the system outages. The optimal fluctuation method, used for analysis of the rare event statistics, is a powerful tool. In this problem it gave us the analytical form of the distribution function of error rates. In general

1 the method gives a description of the region in phase space, which makes the principle contribution to the probability of the event under consideration. Thus, I used the optimal fluctuation to guide the importance sampling for further computational analysis. Importance sampling guided Monte Carlo simulations were performed to evaluate the probability of events occurring with probability ∼ 10−9. Moreover, I computed the distribution of such events, the task that would be impossible to achieve with a direct approach. The results of the analytical and numerical investigations were confirmed using a table-top experiment. I designed and assembled a table top experiment in the lab of Optical Science Center at the University of Arizona. This involved learning the use of high precision equipment, managing the power budget in communication links, developing software for the interface between PC and the data collecting equipment. Finally, I analyzed the collected data and confirmed that the error distribution is log-normal, as was predicted in our analytical study. I presented the results of this work at the SIAM Conference on Nonlinear Waves and Coherent Structures conference in 2006, and later at the 7th AIMS International Conference on Dynamical Systems in 2008. In future I want to apply the analytical and numerical methods used in the above problem to study a related problem. I want to use the method of optimal fluctuations to determine the statistics of the noise induced errors in the communications systems that utilize code division multiple access (CDMA) method of information coding. The two commonly present noise sources make such generalization possible: the presence of amplifier induced temporal noise present in all the systems where information is transmitted over significant distances, and the structural disorder of essential system components, caused by the limited precision of component manufacturing process. Finally, due to the interaction of the disorder system component with its environment, the structural disorder fluctuates leading to strong fluctuations in error probabilities, making system outages unavoidable. I plan to use the method of optimal fluctuations to evaluate the error statistics and develop procedures to avoid system outages.

Solitary wave solutions in Schrodinger¨ equation with periodic dispersion. The leading models describing light propagation in optical fibers are based on the nonlinear Schrödinger equation (NLSE). Current generation of fiber optical communication systems use what is referred to as dispersion management to compensate for the effects of chromatic dispersion, which is modeled as a periodic dispersion in NLSE. It is known that in the presence of weak nonlinearity solitory wave solutions exist. Such solutions propagate through the system with dispersion management as “breathing” pulses, preserving their shape, and therefore can be used as bit-carriers. I investigated bound-pair solitary wave solutions of this system for all system parameters, and found a previously unknown branch of solutions. These types of solutions can be used to increase the transmission capacity of the communication lines by allowing tighter packing of bit-carriers. I am planning to use this method to perform further investigations of solutions, where multiple pulses are bound together, to further improve the transmission capacity. I presented the results of the above work at a mini-symposium at the SIAM Annual Meeting in 2009. I organized the above mini-symposium entitled “ Optical Systems: Nonlinearity and Stochastics.”

2 Neuronal Models and Neuronal Networks A large number of social, technological and biological structures can be described as complex networks. The architecture of these structures can be described by a directed graph, with the dynamical units at the nodes of the graph, which interact with each other by sending pulses via the edges of the graph. An important and challenging task in studying such structures is inferring the architecture of these networks, given their functional properties.

Functional Connectivity of Scale-Free Neuronal Networks. During my position at the Rensse- laer Polytechnic Institute as a Postdoctoral Research Associate, a collaboration with the researchers from Courant Institute was initiated, where my collaborators and I studied functional properties of scale-free neuronal networks. The networks are referred to as scale-free if the probability distribution of node degrees has power-law tails, P (k) ∼ k−γ, with γ satisfying 2 < γ ≤ 3. Such distributions are interesting due to the fact that in the limit of large networks they have a well defined mean, while the second moment diverges. There is experimental evidence for the functional scale-free connectivity of neuronal networks. My collaborators and I investigated whether the scale-free functional connectivity can be inferred from the architectural connectivity. Using a mean-field theory I studied the large-scale, scale-free networks of integrate-and-fire (IF) neurons. I showed that the firing rate in such networks is strongly dependent on the degree- correlation function, i.e. the function that determines the probability for two nodes of given degrees to be connected by a directed edge. This function provides a classification of a large subset of the scale-free networks into two classes, based on whether the mean incoming degree of node’s neighbors is an increasing (assortative) or a decreasing (disassortative) function of the node’s degree. I performed analytical and numerical investigation of the functional connectivity in the two classes of scale-free neuronal networks. In an asymptotic regime of a strong external driving I have shown that the firing rate in disassortative networks is always proportional to the incoming degree of a node. This statement was confirmed with a large scale numerical computation, where a system of ∼ 105 IF equations was solved simultaneously, for 103 network realizations. As a consequence of this statement, I was able to show that the probability distribution of the firing rate indeed has power-law tails. On the other hand, using an example of an assortative network I’ve shown that in these networks the asymptotic behavior of firing rate can be superlinear. In fact, I have shown that the firing rate depends on the incoming degree k as ∼ kα, α> 1. Moreover, I have shown that α depends on the coupling strength between the nodes. I have confirmed this result by the numerical simulations of the scale-free network with 105 nodes. In my future work, I would like to generalize this result for the general assortative networks, with a goal of establishing the conditions when the probability distribution of the firing rates in these networks has power-law tails. Experimental evidence suggests that the neuronal networks are in fact disassortative, that is nodes with larger degree are more likely to be attached to nodes with smaller degree. Moreover, it seems natural to assume that the nodes with large incoming degree also have a large outgoing degree. Thus, as a simple model that takes both of these features into account, I’ve studied disassortative scale-free neuronal networks, where on average incoming degree is equal to the outgoing degree. Analyzing this system, I showed that the firing rate depends on the second moment of the

3 degree distribution, which diverges in the limit of large network size. My analysis provided a way of scaling the coupling between the neurons in the limit of large network size, providing a range of coupling values where one may expect nontrivial behavior of the IF scale-free networks. These results can be utilized for classification of pulse coupled networks based on their functional properties []. On one hand, the tails of the distribution of the firing rates in disassortative networks are independent of the coupling between the neurons, only affecting the mean of the distribution. On the other hand, the effect of changing the coupling in assortative networks is pro- found: the tail of the firing rate distribution may actually change upon the change of the coupling. Furthermore, if disassortativity of the network is an established fact, one can deduce the structure of the degree distribution of the network from its functional properties. I presented this work at a SIAM Conference on Applications of Dynamical Systems in 2009.

Synchrony in Stochastically-Driven Scale-Free Neuronal Networks. Recently, I have worked on a problem of synchronization in stochastically driven, pulse-coupled, integrate-and-fire networks. Specifically, my collaborators and I were interested in synchronization in the scale-free directed networks, as there is strong evidence suggesting that the neuronal networks are in fact scale-free, while synchronous firing is an important behavioral regime of the neuronal networks. Here, the synchronization is characterized by the total firing events, where, as the first neuron fires, it instantaneously brings the voltage of the neighboring neurons closer to the threshold, with some of those neurons crossing the threshold. This process continues until all of the neurons have fired. The problem was solved by solving a first-passage-time problem described by a Fokker-Planck equation. We were able to describe the distribution of neuronal voltages at the time when the first neuron fires. This allowed us to define the condition when the cascades take place. As a result, we computed the firing rate of the system in such synchronous regime. Furthermore, we described the probability of a cascade taking place as a function of the strength of coupling between the neurons in the network. This last problem involved a bulky numerical computation, as well as heavy combinatorial computations, in order to take into account the complexity of the degree correlations and clustering in these networks. Our analysis showed some important similarities and differences in synchronization of scale-free networks and all-to-all coupled networks.

Exponential IF model. The field of bio-physics provides an excellent breadth of problems, among which are challenging problems that can be accessible for the undergraduate students. Dur- ing the academic year of 2008 I have co-advised three undergraduate students on a problem dealing with the neuronal models. Hodgkin-Huxley model is a widely accepted, empirical neuronal model; however, the equations in this model are stiff, making numerical analysis of neurons difficult, and, at the same time, this model has a very large number of parameters, complicating the analysis. The students worked on finding a good approximation to this model with an exponential integrate- and-fire model with adaptation current (aEIF), which is a low dimensional projection with fewer parameters and is much simpler to solve numerically. They have developed two methods for fitting the aEIF to the Hodgkin-Huxley model driven by a stochastic input. The students presented their work at a poster session SIAM Conference on Applications of Dynamical Systems in 2009, and are currently working on publishing the results.

4 In the future I plan to apply the methods of optimal ﬂuctuations to analyze the error statistics in the communication systems based on CDMA information coding. Communications systems utilizing CDMA range from cell phones to ﬁber optical networks, and thus take up a large part of information transmitting market. Thus, this work can lead to a better performing communication systems, and thus have an important impact on the community. Further analysis of the disassortative networks could help lead to a better understanding of the neuronal networks, in particular this could shed light on why this particular arrangement of neurons is natural. On the other hand, better understanding of the assortative networks is necessary to analyze the properties of technological networks such as Internet.