When I tell people that I use mathematics and engineering to study and understand how the brain works, the reaction I get is oftentimes one of confusion. Despite the seemingly head scratching connection between the two, the reality is that we will never be able to understand how the brain works as a system without the use of mathematics and related applied fields of physics and engineering. To understand why, we first need to understand something about how complex the brain is.

The brain is truly a complex system, in the sense that the whole is greater than the sum of its parts.

The brain is complex, not merely complicated

The brain’s incredible capability to learn new things, be creative, and its ability to extrapolate new information from limited and incomplete (noisy) data are the result of an almost uncountable number of internal computations.

To attempt to grasp the size of this computational machine, consider that the brain is a massive network of densely interconnected cells consisting of about 171 trillion brain cells — about 86 billion neurons and another 85 billion non-neuronal cells. There are approximately 10 quadrillion connections between neurons alone — that’s 10 followed by 15 zeros.

Beyond just the sheer size of this huge network, it is important to appreciate that the brain is truly a complex system, in the sense that the whole is greater than the sum of its parts. It has the ability to display hard to grasp emergent properties such as self-awareness and consciousness. Yet even these properties are presumably outcomes of the same brain computations. How it achieves all this remains one of the greatest mysteries in science.

If we are to ever understand all the incredible things the brain can do, we will need to rely on mathematics as a unifying language and framework. We just cannot keep track of everything we know about the brain without it.

For our purposes here, it is important to differentiate complex from merely complicated. The brain is complex in the sense that the individual parts or components that make it up work together to produce emergent properties that cannot be fully explained or understood by a consideration of the interactions of the individual components themselves, such as cells and molecules, even though we have considerable knowledge about cells and molecules and how they function.

In contrast to the brain, consider for example a Space X rocket or NASA space shuttle. Both are clearly very complicated engineering systems, but they are not complex. There is an engineer somewhere that knows the function and role of every bolt and screw, and how that bolt or screw contributes to the operation of the whole system, even if that knowledge is spread across many individuals. However, in a complex system such as the brain this is not the case, and simply knowing everything there is to know about the individual parts does not guarantee an understanding of emergent properties.

Another way to think about the complexity of the brain is by considering its ability to display incredible degrees of plasticity, robustness, fault tolerance, and adaptability. One extreme example of the human brain’s incredible robustness and ability to adapt is a neurological condition called Rasmussen’s encephalitis, a rare pediatric chronic inflammatory neurological disorder that typically affects one hemisphere. It is characterized by severe and frequent seizures that if left untreated result in loss of motor function, loss of speech, cognitive decline, and other neurological deficits. Most patients eventually stop responding to drugs and other medical treatments. In many cases the only effective treatment is a hemispherotomy, whereby portions or the entire affected cortical half of the brain is surgically removed. Or the corpus callosum, the high speed connections between the two halves of the brain, is cut from the unaffected hemisphere. Yet, to varying degrees, the remaining cortex in these patients is able to take up the functions of the lost brain tissue to a remarkable extent. In most cases these patients are able to function cognitively and physically almost normally considering how much of their brains are removed. Now contrast that with what would happen if you removed half the transistors or circuits in your computer.

The role of mathematics in understanding the brain is essential

Given this degree of complexity, if we are to ever understand how all the interactions and computations give rise to the incredible things the human brain (and those of many other specie) can do, we will need to rely on the use of mathematics as a unifying language and framework within which to describe everything. To put it bluntly, we just cannot keep track of all the moving parts, the details of everything we know about the brain, without it.

Neurobiological experiments that generate details about how the brain works at all different scales — genetic, molecular, cellular, and in networks — are at the core of our understanding of a myriad of processes that underlie how the brain works. After all, neuroscience, like all the natural sciences, is an experimental endeavor. However, in the face of such complexity it is simply impossible to arrive at a deep understanding of the brain’s computations and its emergent properties and cognitive capabilities by considering individual details and components in isolation, be they genes, ion channels, proteins, neurons, or entire brain regions. We need the power of mathematics to keep track of all the details and their constantly changing interactions in order to understand how they influence one another. There are simply too many moving parts, too many details. Such mathematical models of the brain are also necessary to allow logical inferences to be drawn that inform our conceptual understanding. How a set of computations A result in an observed outcome E, such as a behavior or other motor output, needs to be logically connected through B, C, and D to infer a causal relationship that gets you from A to E. If the number of dynamically changing details is too great, purely descriptive models that attempt to connect those details ‘in words’ are just not capable enough. We need a mathematical model to keep track of everything.

Of course, as I already stressed, the data and understanding gained by experiments about individual details is critical for informing such mathematical models. The predictive and descriptive power of mathematics when applied to any physical system, including the brain, can only be as good as the experimental data that goes into it in the first place.

It is also important to appreciate that in the course of pursuing a mathematical and systems engineering understanding of the brain, it does not necessarily mean that we have to reverse engineer it in every detail to the point that we are modeling every aspect of how the biology itself implements the brain’s internal algorithms. The substrate, the ‘wetware’, that the brain is made from necessarily constrains how biology is able to execute those algorithms. But the rules and algorithms responsible for brain computations are independent of the wetware. From a mathematical perspective, this means that we can draw a distinction between a level of mathematical detail about what is being modeled in order to capture and describe a brain algorithm, versus the biological details that can be intentionally ignored. In essence, mathematical descriptions abstract away the details of the biological implementation. This opens up the opportunity for a mathematical and engineering understanding of the brain to inform other major engineering fields such as machine learning and artificial intelligence. And conversely, for those fields to inform how we study the brain.

Eugene Wigner (1902–1995) in 1960 wrote a well known essay entitled “The unreasonable effectiveness of mathematics in the natural sciences”. Physics has always taken advantage of this. Computational neuroscience has a similar history in its attempt to understand the brain, with some home runs (such as Hodgkin and Huxley’s Nobel Prize winning work to understand the action potential) but much more limited success overall. Modern neuroscience however is now at an inflection point in the sense that to make truly groundbreaking progress in our understanding of the brain mathematics will need to increasingly play a central role, not a supporting one. God speaks every language with equal fluidity, but his native tongue is mathematics.