# Mathematical Neuroscience

**Control of collective neuronal dynamics**

In the nervous system the physiological synchronization processes are important, for example, in the context of information processing and motor control. However, pathological, excessive synchronization strongly impairs brain function and is a hallmark of several neurological disorders such as Parkinson’s disease, essential tremor, and epilepsy.

Nowadays there is a significant clinical need for effective methods for suppression of undesirable neuronal synchronization. Such control methods are developed in the framework of an interdisciplinary research by combining theoretical and experimental approaches, where the functional neuronal interactions within and between neuronal clusters in the brain can be modeled and investigated.

Methods of nonlinear dynamics and statistical physics are used to model and to understand fundamental mechanisms of neuronal dynamics. We use a modeling approach, which employs mathematical models ranging from ensembles of coupled phase oscillators to generic as well as microscopic neuronal networks, which are used for the designing, testing and calibration of the control techniques. We explore methods to control macroscopic dynamics of neuronal populations in order to contribute to the development of new therapeutic stimulation techniques.

Several methods have been developed with the above approach, which include linear multisite delayed feedback, nonlinear delayed feedback, and proportional-integro-differential feedback for the control of synchronization in either one neuronal ensemble or in several interacting neuronal populations. The developed control methods are also optimized with respect to values of their parameters as well as stimulation protocols.

**Some publications:**

Hauptmann, C.; Popovych, O. & Tass, P. A. “Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study,” *Biol. Cybern.* **93**, 463-470 (2005).

Popovych, O. V.; Hauptmann, C. & Tass, P. A. “Effective desynchronization by nonlinear delayed feedback” *Phys. Rev. Lett.* **94**, 164102 (2005).

Tass, P. A.; Hauptmann, C. & Popovych, O. V. “Development of therapeutic brain stimulation techniques with methods from nonlinear dynamics and statistical physics,” *Int. J. Bif. Chaos* **16**, 1889-1911 (2006).

Popovych, O. V.; Hauptmann, C. & Tass, P. A. “Control of neuronal synchrony by nonlinear delayed feedback,” *Biol. Cybern.* **95**, 69-85 (2006).

Popovych, O. V.; Hauptmann, C. & Tass, P. A. “Desynchronization and decoupling of interacting oscillators by nonlinear delayed feedback,” *Int. J. Bif. Chaos* **16**, 1977-1987 (2006).

Pyragas, K.; Popovych, O. V. & Tass, P. A. “Controlling Synchrony in Oscillatory Networks with a Separate Stimulation-Registration Setup,“ *Europhys. Lett.* **80**, 40002 (2007).

Popovych, O. V. & Tass, P. A. “Synchronization control of interacting oscillatory ensembles by mixed nonlinear delayed feedback,” *Phys. Rev. E* **82**, 026204 (2010).

**Synchronization**

We are interested in dynamical properties of synchronized dynamics and mechanisms of its emergence in networks of interacting oscillators and neurons.

Different models are used to investigate this phenomenon in systems of coupled phase and limit-cycle oscillators as well as in generic neuronal models ranging from a few coupled oscillators to large ensembles.

The impact of system parameters, such as coupling strength, natural frequencies of interacting oscillators as well as time delay in coupling and topology of coupling, on synchronized dynamics is of prime consideration. We investigate the structure of the parameter space and bifurcation transitions leading to the emergence of synchronized dynamics, aiming at the understanding of that how the collective dynamics can effectively be controlled.

Some interesting phenomena have been studied including phase chaos, Cherry flow in coupled phase oscillators, delay-induced chaotic phase synchronization and desynchronization, phenomenon of macroscopic entrainment by an external periodic forcing, phase-locked swallows in parameter space, etc.

**Some publications:**

Maistrenko, Y.; Popovych, O.; Burylko, O. & Tass, P. A. “Mechanism of desynchronization in the finite-dimensional Kuramoto model” *Phys. Rev. Lett.* **93**, 084102 (2004)

Popovych, O. V.; Maistrenko, Y. L. & Tass, P. A. “Phase chaos in coupled oscillators“ *Phys. Rev. E* **71**, 065201(R) (2005)

Maistrenko, Y.; Popovych, O. & Tass, P. A. Chaotic Attractor in the Kuramoto Model *Int. J. Bif. Chaos* **15**, 3457-3466 (2005)

Ashwin, P.; Burylko, O.; Maistrenko, Y. & Popovych, O. “Extreme sensitivity to detuning for globally coupled phase oscillators” *Phys. Rev. Lett.* **96**, 054102 (2006)

Popovych, O. V.; Krachkovskyi, V. & Tass, P. A. Twofold impact of delayed feedback on coupled oscillators *Int. J. Bif. Chaos* **17**, 2517-2530 (2007)

Popovych, O. V.; Krachkovskyi, V. & Tass, P. A. Phase-locking swallows in coupled oscillators with delayed feedback *Phys. Rev. E* **82**, 046203 (2010)

**Spatio-temporal patterns**

We investigate the emergence of regular spatio-temporal patterns and their control in neuronal networks with different coupling topologies (ring, lattice, non-local coupling, etc.) and time delay in coupling. We are also interested in what kind of dynamical structures and (spatio-) temporal dynamics an external stimulation can induce in the stimulated neuronal populations. The stimulation protocols can range from a low-frequency periodic stimulation to coordinated reset stimulation and multisite delayed feedback.

We collaborate with the Research Group "Dynamics and Synchronization of complex systems" (head Dr. Serhiy Yanchuk) of the DFG Research Center MATHEON "Mathematics for Key Technologies," Institute of Mathematics, Humboldt University of Berlin.

**Some publications:**

Hauptmann, C.; Omelchenko, O.; Popovych, O. V.; Maistrenko, Y. & Tass, P. A., “Control of spatially patterned synchrony with multisite delayed feedback,” *Phys. Rev. E* **76**, 066209 (2007).

Barnikol, U. B.; Popovych, O. V.; Hauptmann, C.; Sturm, V.; Freund, H.-J. & Tass, P. A., “Tremor entrainment by patterned low-frequency stimulation,” *Phil. Trans. R. Soc. A* **366**, 3545-3573 (2008).

Perlikowski, P., Yanchuk, S., Popovych, O.V., & Tass, P.A., “Periodic patterns in a ring of delay-coupled oscillators,” *Phys. Rev. E* **82**, 036208 (2010).

Lysyansky, B.; Popovych, O. V. & Tass, P. A., “Multi-frequency activation of neuronal networks by coordinated reset stimulation,” *Interface Focus* ** 1**, 75-85 (2011).