Exploring the Role of Geometry and Combinatorics in Brain Function
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Chapter 1: The Brain's Computational Landscape
The capabilities of the human brain arise from a highly intricate interplay of diverse signaling activities and countless individual computations. Functions such as learning, abstract thought, adaptation, and imagination stem from this extensive combinatorial computational framework. Even complex phenomena like self-awareness and consciousness can presumably be traced back to it. This computational framework is the product of thousands of years of evolution, which is embodied in the brain's physical and chemical substrates, often referred to as its 'wetware'.
The brain can be viewed as a geometric spatial network composed of roughly 86 billion neurons alongside 85 billion non-neuronal cells. There is a significant range of variability within these numbers, as numerous classes and subclasses of both neurons and non-neuronal cells exhibit distinct physiological behaviors and structural differences that influence their dynamics. An astonishing number of signaling events occur at various spatial and temporal scales, interacting and integrating to form the brain's computational complexity and its emergent functions.
Despite its vastness, this computational space is finite, constrained by the physical limitations of the brain's wetware. Recognizing these constraints can enhance our interpretation of experimental data and the development of mathematical models that aim to elucidate brain functions and cognitive processes. One fundamental constraint arises from the relationship between the spatial geometry and the connectivity of neural networks, which regulates the dynamic signaling that neurons use for communication.
Indeed, the significance of this foundational structure-function constraint is so profound that my colleagues Alysson Muotri from UC San Diego and Christopher White from Microsoft Research, and I recently posited that any theoretical or computational model claiming relevance to neuroscience must account for this principle. This holds true even if the model's original intent did not take this constraint into consideration.
Another vital constraint is the energy limitations faced by the brain. The brain has developed various strategies to optimize its energy usage because it consumes a significant portion of the body's total metabolic energy, yet its energy consumption is relatively low—comparable to that of a dim lightbulb. Beyond mere energy limitations, understanding how the brain optimizes its performance under these constraints is crucial for comprehending its design and operational algorithms.
When we grasp how these constraints influence dynamic signaling and information flow across various scales—from individual neurons as mini-networks to collections of neurons and larger brain regions—we can start to view the brain as an engineered system. While this notion is straightforward, the execution is far more complex. Engineers, physicists, neuroscientists, and mathematicians are only beginning to scratch the surface of this multifaceted approach, which drives much of our laboratory's research.
Section 1.1: The Interplay of Structure and Function
The fundamental structure-function constraint emerges from the interaction between anatomical structure and signaling dynamics. Information integration and transmission within the brain depend on the relationship between structural organization and signaling dynamics across different scales. This constraint arises from the geometry—connectivity and path lengths—of the brain's structural networks and the resulting latencies associated with information flow.
For instance, at the cellular network level, individual neurons act as nodes interconnected by axons of varying lengths. Action potentials, which are discrete signaling events, travel between neurons. When these action potentials reach a receiving neuron, they combine until a threshold is met, prompting that neuron to fire and transmit its own action potentials to subsequent neurons. However, action potentials do not travel instantaneously; they propagate at finite speeds influenced by the biophysics of the axonal membranes. The complex geometry of axons creates latencies in the signals carried by action potentials, which subsequently affect the timing of neuron contributions towards reaching the firing threshold.
From a combinatorial standpoint, the timing and summation of action potentials in a given neuron occur independently of other neurons. An adult human brain contains approximately 10 quadrillion connections, with each connection functioning autonomously, largely unaware of its counterparts. Furthermore, neurons may enter a refractory state that prevents them from responding to incoming action potentials, amplifying the impact of timing and latencies on the brain's overall computations. The subtle balance of these competing factors can profoundly affect brain dynamics, as demonstrated by the refraction ratio, which we have shown individual neurons seem to optimize to maintain.
From a broader information-theoretic perspective, these principles apply not only to biological neural networks but also to any physically constructed network. The brain's networks are physical entities through which information must traverse, and the signals encoding this information are subject to processing times and speeds that must cover finite distances to exert their influence.
Section 1.2: The Influence of Geometry on Network Dynamics
The significant impact of the structure-function constraint on network dynamics can be illustrated through numerical simulations of geometric biological neural networks. A disparity in the balance between network geometry and dynamics—specifically, in the refraction ratio—can lead to a complete breakdown in the network's ability to process information, even when the network's structure remains unchanged.
In our numerical experiments, we stimulated a geometric network of 100 neurons for 500 ms, then modified the refraction ratio by adjusting the signaling speed. The outcomes highlighted that at lower signaling speeds, the network exhibited low-frequency periodic activity akin to patterns observed in various brain regions. However, as the signaling speed increased, recurrent activity diminished, ultimately ceasing altogether at the highest speed due to a mismatch in the refraction ratio: signals arrived too quickly for downstream neurons to recover from their refractory periods.
The first video, "Some Geometric Perspectives on Combinatorics: High-Dimensional, Local and Local-to-Global I," discusses how geometric principles apply to combinatorial challenges, providing insights into their relevance to understanding complex networks.
Chapter 2: Toward a Systems Engineering Perspective of the Brain
The ultimate significance of constraint-based modeling in understanding the brain as an engineered system will be realized through its application to experimental data and brain measurements.
In particular, human-derived brain organoids represent a unique experimental model due to their relevance to human brain structure and function. These organoids bridge mechanistic neurobiology with cognitive properties, potentially shedding light on aspects of consciousness. While the direct relevance of organoids to the human brain may vary, other experimental models can be employed to explore specific questions based on their unique characteristics.
For instance, the deterministic and reproducible neurobiological connectivity of the worm C. elegans allows for comparative studies with organoids, while the ability to observe physiological and metabolic properties in intact zebrafish during behavioral experiments offers additional insights. The mathematical frameworks used for modeling across different experimental systems remain agnostic to specific neurobiological details, and constraints like structure-function and energy considerations are universally applicable.
By integrating data across these varied experimental models, rigorous constraint-based mathematical models can provide insights into brain function that no single approach could achieve.
The second video, "Your Brain as Math - Part 1 | Infinite Series," explores the mathematical principles underlying brain function, illustrating how these concepts can enhance our understanding of cognitive processes.