Mathematics to understand brain asymmetry

Many advances in neuroscience came thanks to careful mathematical analysis. For example, Alan Hodgkin and Andrew Huxley used differential equations to describe the action potential, which transmits information between neurons, which earned them the Nobel Prize in 1963. It is now increasingly common to use mathematical tools to analyze huge amounts of data. neural pathways that are being obtained. Beyond this, using fundamental mathematical principles, it has been possible to support conjectures about the functioning of the brain that have been put forward for decades or centuries.

These are relevant hypotheses with broad support in the community, despite adding little empirical evidence or the fact that there are still no definitive mathematical arguments. They are reminiscent of famous conjectures—such as Fermat's last theorem or the Riemann hypothesis—that challenge several generations in search of rigorous confirmation.

An example is the predictive brain hypothesis: much of our cognitive abilities result from an evolutionary pressure to anticipate our environment. The great success of the calls large language models (LLM), such as OpenAI's ChatGPT or Meta's LlaMa, supports this idea. Their spectacular “intelligence” results from a single task: maximizing the probability of getting the next word right, given an incomplete text. This shows that prediction allows the development of advanced cognition. It remains to be known whether, in addition, cognitive abilities always require this type of skill.

Another example that has just gained mathematical support is the hypothesis that greater cognitive complexity leads to lateralization, or breaking the mirror symmetry of the brain. This idea has been accepted almost since the beginning of neuroscience (reaching textbooks and popular science books). More than 150 years ago, the discovery of Broca's area (responsible for language generation) confirmed two relevant facts about the brain: that there is localization (that is, different regions implement different tasks) and that it is asymmetric, since this area It is usually located in the left hemisphere. This led to the postulation that lateralization results from advanced human intelligence. Over time, examples of asymmetry were discovered in many other species, softening the hypothesis: greater cognitive complexity exerts evolutionary pressure towards brain lateralization.

Until recently this was only supported by anecdotal observations. More empirical evidence is difficult to obtain due to the challenge of measuring brain asymmetry and quantifying cognitive complexity. Furthermore, there was no adequate theoretical framework that would allow us to pose—not just answer—the conjecture in a rigorous manner.

A new work published in Physical Review offers this framework and provides a robust mathematical argument that supports the hypothesis. To do this, a mathematical model is used, inspired by the science of complex systems, in which neural modules and circuits are reduced to abstract units. These units encapsulate the complexity of emerging cognition, the probability that neural circuits make errors, and the costs of using these circuits, such as the metabolic expense of coordinating both hemispheres, or the energy dissipated by any irreversible operation.

The key to the mathematical argument lies in a conflict between the so-called logical operators, mathematical expressions whose result is a Boolean value (true or false). We start from a simple cognitive task, which can be solved with an irreducible neural circuit—that is, it cannot be decomposed into subtasks. The brain could use a single copy of this circuit, located in one hemisphere or the other (to which the logical operator is assigned XOR); use two coordinated copies of the same circuit, each located in a hemisphere (to which the expression AND); or some intermediate combination (OR). The first option is cheaper, but the second may be more robust against neuronal failures.

With all possible configurations, a utilitarian calculation is carried out that takes into account costs and benefits. This allows a map to be created that details when lateralized solutions are preferred over symmetrical ones based on the model parameters, which are costs and an error rate. A first result is that there are no intermediate configurations: there will always be bilaterality or a total break in symmetry.

A second result, and with it the solution to the central hypothesis, appears when considering complex cognitive tasks. In the model, these consist of various subtasks and require composite circuits to be implemented. This results in an operation AND recursive: to benefit from advanced cognition it is necessary to implement without error a subtask, and another, and another, etc. By introducing this new operator in the utilitarian calculation, the map is altered: regions that previously demanded bilaterality begin to prefer lateralization, forcefully showing the existence of evolutionary pressures to lose brain symmetry as cognitive complexity increases.

A region also appears that prefers lateralization for simple tasks, but requires duplicate circuits for complex tasks. Thus, cognitive complexity can promote the evolution of new redundancies, functioning as an evolutionary engine that generates or breaks symmetries in complementary circumstances. The mathematical framework indicates when each possibility will occur, according to the metabolic expenditure of the neuronal substrate, its error rate and the complexity of the contemplated task.

These mathematical conditions, and the inescapable structure of certain abstract objects, constrain physical reality and the biological materialization of our cognitive abilities.

Taking mathematics seriously, it is possible to constrain what designs are possible and probable on a neural substrate and the corresponding mental representations. And, as more empirical evidence arrives, this central hypothesis of neuroscience now rests on a robust analytical framework.

Luis F Seoane is a researcher of Higher Council of Scientific Research at the National Center for Biotechnology and part of Interdisciplinary Group of Complex Systems of Madrid

Agate Timon G Longoria She is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT)

Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician
Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”

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