Math to understand brain asymmetry

Many breakthroughs in neuroscience have been achieved through meticulous mathematical analysis. For instance, Alan Hodgkin and Andrew Huxley used differential equations to explain the action potential, which is responsible for transmitting information between neurons. Their remarkable work earned them the Nobel Prize in 1963. Nowadays, it is increasingly common to employ mathematical tools for analyzing vast amounts of data regarding neural pathways. Additionally, by applying fundamental mathematical principles, it’s now possible to support hypotheses about the brain’s functioning that have been proposed for decades or even centuries.

These hypotheses have wide support in the community, but lack empirical evidence or definitive mathematical arguments. They resemble famous hypotheses like Fermat’s last theorem or the Riemann hypothesis, which have challenged generations in the pursuit of rigorous confirmation.

One example is the predictive brain hypothesis, which states that our cognitive abilities are largely influenced by the need to anticipate our environment. The impressive success of large language models (LLM) like OpenAI’s ChatGPT or Meta’s LlaMa supports this idea. These models exhibit remarkable “intelligence” because they excel at predicting the next word in a given text. This demonstrates how prediction plays a crucial role in the development of advanced cognition. However, it’s still uncertain whether cognitive abilities always rely on this particular skill.

There is growing mathematical support for the hypothesis that greater cognitive complexity leads to brain lateralization, or it disrupts mirror symmetry. This idea has been widely accepted in neuroscience for over 150 years, ever since the discovery of Broca’s area, which is responsible for language generation. This discovery confirmed two important facts about the brain: different regions perform different tasks, and the brain is asymmetric, with Broca’s area typically located in the left hemisphere. This led to the suggestion that lateralization is a result of advanced human intelligence. As more research was conducted, examples of asymmetry were found in many other species, suggesting that increased cognitive complexity exerts evolutionary pressure on brain lateralization.

Until recently, gathering empirical evidence to support this was challenging due to the difficulty of measuring brain asymmetry and quantifying cognitive complexity. Additionally, there was no adequate theoretical framework that would enable us to pose, rather than just answer, the hypothesis in a rigorous manner.

A new paper published in Physical Review X presents a framework and a strong mathematical argument to support the hypothesis. The study utilizes a mathematical model inspired by the science of complex systems, simplifying neural modules and circuits into abstract units. These units capture the complexity of cognition, the likelihood of circuit errors, and the associated costs, such as the metabolic expense of coordinating both hemispheres or the energy dissipated by irreversible operations.

The key to the mathematical argument lies in a conflict between logical operators, which are mathematical expressions that produce either true or false values. We begin with a simple cognitive task that can only be solved by an irreducible neural circuit, meaning it cannot be broken down into smaller subtasks. The brain has a few options: use a single copy of this circuit in one hemisphere (the logical operator XOR — exclusive OR); use two coordinated copies of the circuit in each hemisphere (logical operator AND); or find some combination in between (inclusive OR). The first option is more cost-effective, while the second option may offer better resilience against neuronal failures.

To determine the optimal configurations, a utilitarian calculation considers both costs and benefits. This calculation produces a brain map that outlines when lateralized solutions are favored over symmetrical ones based on the model parameters, which include costs and an error rate. Notably, there are no intermediate configurations; the outcome will always be either bilaterality or a complete break in symmetry.

Considering complex cognitive tasks reveals a second result and provides a solution to the central hypothesis. These tasks involve multiple subtasks and require the implementation of composite circuits. This implies a recursive AND operation: to achieve advanced cognition, one must successfully execute each subtask in succession. Introducing this new operator alters the utilitarian calculation, causing regions that previously favored bilaterality to prefer lateralization. This demonstrates the existence of evolutionary pressures to reduce brain symmetry as cognitive complexity grows.

A region emerges that favors lateralization for simple tasks but needs duplicate circuits for complex tasks. As a result, cognitive complexity can drive the development of new redundancies, acting as an evolutionary force that creates or disrupts symmetries in appropriate situations. The mathematical framework predicts the occurrence of each possibility based on the metabolic cost of the neural substrate, its error rate, and the complexity of the task at hand.

Mathematical conditions and the inherent structure of abstract objects limit physical reality and influence the biological manifestation of our cognitive abilities. By applying mathematics rigorously, we can limit the range of designs that are feasible and likely on a neural substrate and the corresponding mental representations. Furthermore, as more empirical evidence emerges, this core hypothesis of neuroscience can now be supported by a robust analytical framework.

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