Dynamical Systems in Brain Science, why bother?

by Hiba Sheheitli

“I would say the beginning is 1666, Isaac Newton is at that point 24 years old, and he has a big year in 1666, more or less invents calculus, figures out the laws of optics, discovers universal gravitation, it’s a good year for him, he has other results.. but anyway, as far as our concerns, I mean this is not really true to say that he invented differential equations, but he goes much further with them than anybody else up to that point, and, for the first time in humanity’s history, explains the orbits of the planets, that is, Kepler has already calculated from Tycho Brahe’s data that the planets move in ellipses along with the other two laws of planetary motion and Newton can explain all of them from universal gravitation plus calculus plus his three laws of motion. So, that is the beginning of dynamics, but then, interestingly, he has solved at that point what we think of as the two-body problem: earth or another planet being pulled by the sun’s gravity, that’s when he is able to explain elliptical orbits. To do that, he had to neglect all other planets in the solar system or all other objects in the universe, so that’s why we call it the two-body problem. But what if there were more bodies? Like if there’s the moon or other planets, well, that’s the three-body problem or the N-body problem, and Newton can’t solve it, and he actually writes to one of his friends that no problem has made his head ache like the problem of the three bodies. So, he does not solve it, neither does anybody else, but people work on it for several hundred years, fast forward to around the late 1800’s, it’s 1890 by now and still no body has solved the three-body problem, everybody has tried, Euler, all the greats, Gauss… and finally, Poincaré explains what the trouble is, in fact, you can’t really solve the three-body problem. So, Poincaré introduces a geometric approach, as opposed to the analytical calculus-based approach… he uses geometry and visualization to supplement calculus and sees the thing that we now call “chaos” for the first time.” Chaos, in this mathematical sense, can be understood as aperiodic, seemingly unpredictable, behavior in a deterministic system, displaying sensitive dependence on initial conditions.” [1]

As such, the story of the birth of the dynamical systems school of thought is very telling of the powerful role it plays in efforts of understanding complex behavior of nonlinear systems. While Newton knew the exact physical laws governing the motion of planetary bodies, and was able to express those laws as ordinary differential equations, the nonlinear and high-dimensional nature of those equations stood in the way of cracking the behavior of the system; he was unable to obtain specific solutions to describe the motion of the planets given specific parameters and initial conditions. Poincaré’s alternative geometric approach was to, instead, ask a different question: what does the space of all possible solutions look like? So, imagine one could represent and visualize the state of the system as a point in an abstract mathematical space, which will be called state space (or phase space), in which the coordinates of that point will be the value of the interacting dependent variables that will vary in time. Then, the behavior of the system can be visualized as the trajectory that such a point will trace in this state space as the system evolves in time. The relevant question then becomes: what does the portrait of all such possible trajectories look like? The aggregation of all possible solutions of the system can then be thought of as a “flow” that tells us all there is to know about the global dynamics. Think of a bathroom sink, if you wanted to understand the flow of water in it, it is enough to localize the faucets and the drains, no need to follow the individual water molecules, all is repelled away from faucets and attracted into drains.

Such is the case in abstract phase space, the global flow of the dynamics will be governed by the existence or absence of repellents and attractors, such as equilibrium points, limit cycles (periodic solutions), strange chaotic attractors, manifolds or other geometric creatures that can live in higher dimensions. The next interesting question would be: how does the number, nature and stability of these latter repellants and attractors depend on system parameters? What bifurcations are possible? Bifurcations being qualitative changes that the flow in phase space can undergo as system parameters are varied, for example, changes in the number or stability of equilibrium points. We can also ask, what is the basin of attraction for each attractor? That is, the set of initial conditions that leads the system to corresponding attracting states. So, it is the big picture of what is happening, the global behavior of our system, that is more of interest and significance than a single instance or trajectory. Dynamical systems theory sets the framework and offers the tools for tackling such questions.

In brain science, the first footprints of dynamical systems came with the groundbreaking famous Hodgkin-Huxley model[2] describing the generation and propagation of action potentials in neuronal axons and taking the form of a system of nonlinear differential equations. Serving as the cornerstone for the ever growing biophysical knowledge of neuronal dynamics on the cellular level, it opened the door for a plethora of dynamical systems tools[3] to be deployed for the representation and characterization of the basic features of neuronal dynamics[4], such as, excitability and threshold behavior, bursting oscillations, synchronization, multistability, hysteresis and spontaneous multiscale spatiotemporal oscillations and organization; dynamical features that are already observed and well understood in numerous other fields[5] and that can be probed and exploited in the study of the less well understood neuronal systems. In this process, a conceptual bridge is built across different fields of study through an underlying universal mathematical milieu that is the machinery of dynamical systems.

Building on such efforts, it is possible to attempt to construct biophysically faithful mathematical models for whole brain dynamics that incorporate individual neuron models. However, with billions of neurons making up a living brain, such models would consist of an extremely high-dimensional system of coupled nonlinear differential equations and would thus pose a somewhat similar difficulty to that faced by Newton in his N-body problem. Admittedly, advancements in high-performance computing, which Newton lacked in his days, enhances the feasibility of simulating such complex nonlinear high dimensional mathematical models[6], given that we choose specific parameters and initial values. Then, why bother with dynamical systems in brain science?

 Just like the laws of mechanics were necessary but not enough for Newton to conquer the three-body problem, a more global characterization, and a deeper understanding of the fundamental nature of whole brain dynamics, calls here for the equivalent of Poincaré’s geometric approach to Newton’s N-body problem. An approach that guides the extraction of essential dynamical ingredients starting from the microscale but zooming outwards towards the meso or macro scale, to arrive at representative canonical mathematical models that elucidate core mechanisms behind the rich dynamic repertoire of brain activities[7]  and provide more computationally efficient alternative building blocks for whole brain models[8].

 Ultimately, we can write down equations for the highly nonlinear, high-dimensional, complex system that is the brain, and then, a most pressing challenge presents itself: how to explain the enigma of the emergence of order, ‘’mind’’ and ‘’behavior’’, out of all that bewildering complexity; that order that manifests as intricate low-dimensional spatiotemporal processes observed in neural signal measurements[9]. Here, dynamical systems theory provides the landscape, jargon and machinery to address this challenge, to construct much needed guiding conceptual frameworks[10] that aim at demystifying the mechanics of the dynamical echoes and shadows of our inner lives.

To abstract away from experimental data, of various forms and spatio-temporal scales, into mathematical state space where dynamical systems tools can be wielded to unfold what is not observed but yet governs the observable; herein lies the power of this school of thought. It is an approach that might seem like a feat of artistry but is better viewed as elegant craftmanship that has served science, rigorously, across its various fields and is, undoubtedly, indispensable for its last frontier, the brain.

[1] Steven Strogatz, introductory lecture to his course on nonlinear dynamics & chaos:

https://www.youtube.com/watch?v=ycJEoqmQvwg&list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V

[2] Hodgkin AL and Huxley AF. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve." J Physiol. Aug;117(4):500-44. https://doi.org/10.1113/jphysiol.1952.sp004764

[3] Izhikevich, E. M. (2007). Dynamical systems in neuroscience. MIT press.

[4] Rinzel J. and Ermentrout G. B. (1989) Analysis of neural excitability and oscillations. In Koch C., Segev I. (eds) Methods in Neuronal Modeling, Cambridge, Mass: MIT Press.

[5] Strogatz, S. H. (2018). Nonlinear dynamics and chaos with student solutions manual: With applications to physics, biology, chemistry, and engineering. CRC press.

[6] https://ebrains.eu/service/the-virtual-brain/

[7] Jirsa, V. K., Stacey, W. C., Quilichini, P. P., Ivanov, A. I., & Bernard, C. (2014). On the nature of seizure dynamics. Brain, 137(8), 2210-2230. https://doi.org/10.1093/brain/awu133

[8] Stefanescu, R. A., & Jirsa, V. K. (2008). A low dimensional description of globally coupled heterogeneous neural networks of excitatory and inhibitory neurons. PLoS Comput Biol, 4(11), e1000219. https://doi.org/10.1371/journal.pcbi.1000219

[9] Mehrkanoon, S., Breakspear, M., & Boonstra, T. W. (2014). Low-dimensional dynamics of resting-state cortical activity. Brain topography, 27(3), 338-352. https://doi.org/10.1007/s10548-013-0319-5

[10] Jirsa, V. (2020). Structured Flows on Manifolds as guiding concepts in brain science. In Selbstorganisation–ein Paradigma für die Humanwissenschaften (pp. 89-102). Springer, Wiesbaden. https://hal.archives-ouvertes.fr/hal-02527445/document