On the general theory of neuronal dynamics
Creators: name: Vinci, Gianni Valerio
Publisher: name: Zenodo
Year: 2023

description: The brain is arguably the most complex system known to man. Under the eyes of a physicist, brains show all the major complications that one can think of when it comes to building a theoretical description. At all scales, the brain is out-of-equilibrium, as any living system must be to avoid death. Moreover, there is no such thing as two truly identical neurons: heterogeneity is a fundamental element of the puzzle. A clear distinction between environment and system is only partially possible. This is due to the intricate feedback loops necessary to adapt and respond to what happens around us. Finally, neurons are stochastic and non-linear systems which certainly does not simplify the picture. On top of that, we have to consider a complication that is common to most scientific endeavors: the system is observed through a keyhole, only partial information is available or, in other terms, we have access in experiments to only a limited number of degrees of freedom. Under these premises, it is natural to wonder whether it is even possible to tackle these problems with a reductionist approach let alone a first-principle reasoning. The unreasonable effectiveness of a mathematical description of the world is however surprisingly universal and it extended beyond traditional physics. The fil rouge of this work is precisely that, despite numerous difficulties, it is indeed possible to gain significant insights into the inner workings of neural systems adopting the powerful toolbox of physics. Understanding the physics supported by networks of neurons is a necessary step in investigating some of the most outstanding mysteries of Nature such as memory, language, and consciousness. Physicists and mathematicians have had an interest in neuroscience for a long time. For example, it is difficult to underestimate key conceptual contributions by scientists of the caliber of Von Neuman and Turing. A more recent example dates back to some decades ago when a burst of research originated in the field of statistical mechanics starting with the seminal work of Amari and Hopfield that lay out the basis of attractor neural network theory. Back then the idea of storage of memories as an energy landscape was purely based on theoretical reasoning but it has now found a large consensus thanks to numerous experimental findings that support it [KF22]. Today we are living in a golden age of neuroscience thanks to unprecedented technological advances. It is now possible to trace and stimulate single specific neurons in behaving animals. Vast networks have been mapped out, down to the level of counting every single synapse in a cube milliliter of cortical surface. Consequently, the need for a unifying theoretical framework has become more pressing. This thesis is an effort in that direction. Following the tradition of statistical physics, I will derive a general theory of neuronal population dynamics starting from first-principle i.e. relating single neuron features to the emerging neuronal network behaviour. Our starting point will be a mean-field theory commonly referred to as the population density approach, that rests mainly in assuming an infinite number of neurons. This kind of theories were developed when it became clear that neurons process most of the information not alone but at a network level. A point of view highly at odds with the "neuron doctrine" put forward by some of the father of the neuroscience like Cayal, Adrian and Eccles. The paradigm shift to a population coding view justified a massive reduction of complexity, since the interest was non longer on single neurons but on the average activity of a circuit. Among the major achievements of this view there is for example the concept of balance state: a balance of excitation and inhibition is necessary to sustain irregular the kind of asynchronous activity which is commonly observed in experiments . Most of the theoretical results refer to the statistics of stationary states or linerizible regimes. While those achievements fostered profound insights, neural circuits are constantly bombarded by external stimuli. Consequently, the transient dynamic and dynamic response are equally important and most open questions orbit around the dynamical non-stationary properties of neural populations. In this work, we will concentrate on three main open questions, all deeply interconnected, regarding neuronal population dynamics. Universality, low dimensionality, and finite-size effects. Thermodynamics taught us that despite the original large number of degrees of freedom the law that governs the evolution of mesoscopic quantities is often low dimensional. The same applies to neural populations, in fact, phenomenological models with few degrees of freedom, of neural activity have been proposed and used substantially in the literature. Much fewer attempts were made in deriving such laws from first principles. The main obstacle in this path is the non-linear nature of the equations. To solve this moment-closure problem, several authors like made ansatzs that inevitably limit the applicability of the resulting theory to regimes that are not always clearly defined. An open question that will address here is then how to derive such a low dimensional theory valid in the most general case and what it tells us about neuronal dynamics. The complexity inherent to biology led in the years to a proliferation of different models for the behavior of single neurons. The main discriminant is the level of detail included in the description, ranging from point processes such as the Integrate-and-Fire framework to the biologically realistic Hodgkin and Huxley model. When we look at the emergent behavior of such different models we find surprising similarities. In a way we could say that the large-scale neuronal dynamics appear to be practically universal, meaning that are weakly dependent on specific features or parameters and more related to the general mechanism at play. The sloppiness of models has been related to emergent, or effective, theories in different contexts but, in neuroscience, the origin of this phenomenon has not been studied in detail and many questions remain to be answered. We will focus here on two: what are the essential ingredients of single neurons that will lead to excepted mesoscopic dynamics? Are all the parameters of the models uniquely determined by the observed activity? Finally, cortical neural networks consist of a finite, albeit huge, number of neurons. It is common knowledge that finite-size impacts the predictions of mean-field theories. The main effect of a finite number of neurons is that mean activity alone is not enough to correctly describe the time evolution of the network. A common approach is then to go beyond mean-field theory and try to capture also endogenous fluctuations generated by finite-size effects. In weakly coupled networks we can use linear response theory, a tool, however, that fails close to bifurcations or in the non-linear regime. Coincidentally it is precisely in the non-linear cases that finite-size fluctuations are most prominent. A good example is the meta-stable activity regime commonly observed in neural recordings where finite-size fluctuations induce transitions between different states and thus are an essential element in an accurate description of neruonal activity. Except for some remarkable results a general self-consistent approach to include these effects in an extended mean-field theory is still lacking.