The Scientist Who Unriddled the Mathematics of Mind: Stephen Grossberg

An introduction to the second greatest scientist in the history of the world, whose math revealed the path to explaining autism.

Whenever you claim to be "the first to do" this or that in artificial intelligence, it is customary—and correct—to add "with the exception of Stephen Grossberg." Quite simply, Stephen is a living giant and foundational architect of the field. 
   .Karl J. Friston, neuroscientist

This article is adapted from an excerpt from Journey of the Mind.

1.

From a very early age, Isaac Newton was fascinated by how things moved. The way a ball soared through the air or plummeted to earth. The way shadows swelled and shrank. How a breeze tossed a leaf hither and yon. This youthful passion instilled within Newton an abiding interest in the dynamics of matter. In a word, he was fascinated by change.

Newton spent his childhood fashioning sundials and windmills. He was particularly enchanted with kites, which rose and dipped in the wind. He experimented with new shapes and sizes, always seeking to improve upon his designs. One reason Newton preferred mental pursuits over physical contests was he was smaller and frailer than his classmates. When unavoidably pressed into athletic competition, Newton would wield what he called “philosophical thinking.” Once there was a contest among the schoolboys to see who could leap the farthest. Newton came in first place, defeating his taller and stronger opponents. His trick: timing his jumps to coincide with advantageous gusts of wind.

When he entered Trinity College at Cambridge, his youthful passions collided with an unexpected obstacle. At the time, English universities embraced Aristotle and the belief the sun revolved around the Earth. Scientific inquiry at Cambridge, such as it was, employed little math, instead relying on qualitative descriptions of objects and events. Students were taught that one set of rules governed things on Earth and another set of rules governed things in the heavens—that there existed a celestial mathematics and a terrestrial mathematics that did not overlap. There was little in such academic dogma to attract Newton’s burgeoning fascination with the dynamics of matter. Instead of attending to his schoolwork, Newton searched for inspiration outside the curriculum.

He read Copernicus and Kepler, freethinking astronomers who broke free of the Late Middle Ages supermind to advocate for heliocentrism. He read Galileo, who proposed a new vision of the dynamics of matter based upon inertia. He was particularly excited by Descartes, who formulated a highly controversial conception of nature as an intricate and impersonal machine, a notion that Newton found congruent with Democritus, who believed that reality was formed out of the constant motion of matter.

Out of dynamics.

Intoxicated by these unconventional ideas, Newton pursued his own private and idiosyncratic studies. He began to see the world as a collection of ever-changing patterns, rejecting the orthodox view that the universe consisted of static objects. Newton speculated that these ever-changing patterns did not represent random chaos but might instead represent a hidden form of order. He realized that if he wanted to apprehend this hidden order, he would need to create new mental tools to support his subversive intuitions. Most especially, he would need to create a new kind of mathematics.

Just then, an unbidden opportunity rolled onto Newton’s doorstep. A pandemic. The bubonic plague began tearing its way through seventeenth-century England. Trinity College went on lockdown. Newton returned home and created a home office to continue his eccentric studies in isolation. Divorced from his professors and classmates, he began constructing models of the motion of objects. These models generated new mathematical ideas that he tested through homemade experiments or endless, obsessive calculations.

Time quickly became a fixation. Newton came to view time as a steady, infinite flow that provided the essential framework for all movement. He devised new mathematical concepts that would enable him to model activity and change. One represented a “flowing quantity.” He called this a “fluent.” Another represented a rate of change. He called this a “fluxion.” By the time he was twenty-three, he had derived an entirely new mathematics of fluents and fluxions. This unprecedented math allowed him to make accurate predictions about the way things emerged, transformed, and vanished. Today we call it calculus.

By late 1666, a tumultuous year scarred by Black Death and the Great Fire of London, young Isaac Newton, working alone and incommunicado, became the most advanced mathematician in the world. He spent two decades consumed with applying his revolutionary mathematical tools to the study of motion. Finally, in 1687, he published Principia Mathematica. It contained the first unified theory of the dynamics of matter. A groundbreaking unity knitting up the cosmos, for Newton’s audacious claim was that all physical objects existing anywhere in the universe were all governed by the same rules.

For the first time in humankind’s conception, heaven and Earth were conjoined within a single seamless system—a mathematical system that obeyed precise and inviolable equations.

Physics boasts a history replete with prodigies, individuals who flashed brilliance young before going on to make extraordinary contributions to the field. Enrico Fermi, one of the lead physicists on the Manhattan Project, wrote a doctoral-level paper on vibrations at the age of seventeen. James Maxwell, the founder of the field of electromagnetism, published his first scientific paper at age fourteen. But towering over them all looms Isaac Newton, humankind’s first mathematical physicist, who as a student intuited that the prosaic behavior of stones, sticks, and bricks followed the same principles as the celestial behavior of moons, planets, and suns.

Things are different in mindscience, where prodigies are scarce. Few eminent psychologists and neuroscientists demonstrated a precocious talent for fathoming Mind as adolescents. There is one notable exception to this rule, however. A mathematical pioneer whose story, in many ways, parallels that of Newton: Boston University professor emeritus Stephen Grossberg.

2.

Grossberg was born in New York City in 1939, the grandson of Hungarian immigrants. His mother was a schoolteacher. His stepfather, an accountant. (His biological father died when he was one.) He was raised in Jackson Heights, Queens, a lower-middle-class neighborhood clamoring with fiercely competitive Jewish boys. From a young age, Grossberg was keenly aware of the fact that living things are either growing or dying, blooming or decaying. This instilled within him an abiding interest in the dynamics of life. In a word, he was fascinated by change.

Even as a child, Grossberg was consumed by the realization that we are only temporary visitors upon this earth. This awareness provoked an urgency to make contact with something more durable than his own ephemeral life. “This, to me, was a deeply religious feeling: how to be in touch with the enduring beauty of the world, even though you can only personally be here for a very short time,” Grossberg says. “I wanted to do something where I could touch the eternal. It seemed the only way to do that at the time, given my limited options because my parents had no money, was to be incredibly good in school.”

Grossberg battled his way to the top of his classes at the elite Stuyvesant High School in Manhattan. He nailed perfect scores on his SATs. He was rewarded with his choice of universities and decided to attend Dartmouth College, which offered a program permitting him to spend his senior year focusing exclusively on research. . . though he possessed only the vaguest idea of what research actually was.

In his freshman year, he took a required introductory course in psychology. It was his first exposure to the science of Mind. “The class unexpectedly created a storm of ideas inside my head,” Grossberg recounts. “I was entranced by the implications of the data on human and animal learning for how things were going on moment by moment inside our minds, the dynamics of minds.”

In the psychology class, the young man was introduced to an unsolved mystery that captivated him. The mystery was known as the serial position effect. First discovered by German psychologist Hermann Ebbinghaus in the 1880s, it had baffled scientists for almost a century. The serial position effect describes how we remember a list of items, such as a grocery list, when items are presented to us one at a time: milk, sugar, potatoes, apples, tomatoes, bacon, yogurt, squash. The serial position effect refers to the fact that people tend to recall the first items in the list as well as the items at the end of the list, no matter how long the list. Items in the middle are usually forgotten. The unanswered question, Why?

Grossberg’s teenage attempt at solving this puzzle altered the course of his life—and the course of the science of Mind. It would open the door to solving consciousness, free will, self-consciousness, the fundamental nature of reality—and the neural dynamics of the dark gift.

3.

Grossberg attended Dartmouth in the late 1950s. Buddy Holly, the Big Bopper, and Ritchie Valens had not yet boarded their fatal plane ride. Hawaii was not yet a state. Ford was still manufacturing the Edsel. In the United States, two antagonistic disciplines held sway over mindscience: psychoanalysis and behaviorism.

Freud’s pseudoscientific theories of the unconscious were defiantly unempirical, qualitative, and mathless. They held that thinking emerged (somehow) from the conflict between an ill-defined ego (the conscious self), superego (a kind of Jiminy Cricket–like conscience), and id (selfish cravings unfettered by morality). Behaviorism, characterized by pigeons pecking at lights and mice dashing through mazes, was ruthlessly empirical and quantitative and, most significantly, rejected any model of thinking. Behaviorism’s defining assumption was that you could understand everything about an organism’s mind by focusing on the statistical relationships between sensory inputs and behavioral responses without speculating about “intervening variables,” behaviorists’ dismissive term for thinking, perceiving, and feeling.

The psychoanalysts said, We don’t need no stinkin’ math to understand thinking!

The behaviorists said, We can understand thinking using simple math relating inputs to outputs!

In any event, neither psychoanalysis nor behaviorism could account for the serial position effect. At all. Its peculiar shape didn’t match any theory of Mind. The mystery enthralled Grossberg.

“One of the most interesting things about the serial position effect was that it seemed to involve learning that could go forward and backward in time,” Grossberg explains. “Items that arrived after the beginning of the list were forgotten, but items that arrived before the end of the list were somehow retroactively forgotten, too. The idea that events could go backward in time fascinated me.”

Even after the psychology course ended, he spent his sophomore year wrestling with the serial position problem, convinced it must hold the key to unlocking well-guarded secrets of Mind. He devised new mathematical concepts that would enable him to model activity and change. He invented representations for mental states, mental events, and the interactions between events and states over time. This is exactly what Newton had accomplished three centuries earlier: Newton pieced together first principles of a whole new field based upon an assiduous mathematical consideration of the dynamics of matter, while Grossberg pieced together first principles of a whole new field based upon an assiduous mathematical consideration of the dynamics of Mind. (In Grossberg’s case, without ever needing to refer to the dynamics of matter, just as Newton had no need of the dynamics of purpose. )

Grossberg’s solitary struggles led him to a revolutionary perspective on the nature of thinking. On the nature of purpose in the cosmos.

One of the groundbreaking features of this new perspective was Grossberg’s realization that any explanation of the serial position effect would need to describe its operation in real time. That is, the explanation would have to account for how the brain changed moment by moment as each new item was presented. More generally, it was apparent to Grossberg that a sophisticated treatment of time itself would be a key factor in any explanation of thinking. Time was not a neutral marker of the passing of mental events nor was it a steady Newtonian flow providing a framework for biological thought. Instead, time was stitched into the very fabric of Mind. Minds were wholly designed from the ground up to manipulate and exploit the flow of time.

This is the crux of the Dynamic Mind perspective: all thinking is activity. The word “mind” is an action noun, like explosion, freefall, or whirlwind.

Building on these insights, Grossberg deduced there must be two distinct but codependent mental processes underlying the serial position effect, each associated with a different rate of time. The first was a fast short-term memory process that changed in real time as new items were presented, and the second was a slow long-term memory process that acted on a much slower time scale and was influenced by the results of the fast activity.

One useful way to think about these distinctive mental processes is to consider ocean waves crashing on a sandy beach. The waves splashing onto shore are like a mind’s fast, short-term dynamics. They flow and recede in real-time, as we watch. The sand on the beach is like a mind’s slow, long-term dynamics. The exact shape and distribution of the sand on the beach, including the curved lines of sand (the swash lines) that mark the highest place each wave reached, are influenced by the real-time activity of the waves, yet retain their physical structure even after the waves recede (say, at low tide). At the same time, the precise distribution of sand on the beach influences the activity of each subsequent wave that interacts with the beach.

A monster breakthrough in mindscience, and I would argue the commencement of the centuries-delayed mathematization of Mind: the notion that thinking consists of multiple mental activities operating on different time scales, yet these activities directly interacted with one another to produce a holistic (mind-wide) behavioral response.

Next came perhaps Grossberg’s most original leap of intuition, an idea grounded in a concept first introduced in Darwin’s theory of evolution, which was humankind’s first foray into the dynamics of purpose. Regarding the items in a grocery list, Grossberg realized there must be some kind of competition between the representations of each item as they were presented to the mind. As if the words “milk” and “sugar” and “banana” battled one another for their rightful slot on the mind’s grocery list. Grossberg discovered that this winner-take-all dynamic was necessary to represent a sequential list in the mind. (Years later, he would demonstrate that winner-take-all dynamics also prevented the degradation of mental lists due to the incessant biological “noise” generated by cellular activity¹.)

From these insights, Grossberg derived a novel mathematical approach to modeling thinking by the end of his junior year. He had become the first mathematical mindscientist, as Newton had become the first mathematical physicist. Grossberg concluded that the best way to make sense of the mind was by representing it as interacting forms of goal-directed activity characterized by nonlinear differential equations operating in real time.

For Grossberg, Mind was made of the dynamics of purpose.

4.

There was every reason to suspect that Grossberg was traveling down an intellectual dead end. After all, he was an adolescent pursuing his own private and idiosyncratic studies without any connection to what any other scientists were doing. A few years later, when he shared his mathematical models with the prominent neuroscientists of the era, the most common reactions were confusion and apathy².

Indeed, over the next few decades many approaches to Mind would come online markedly different from the Dynamic Mind perspective Grossberg was developing, including two perspectives that dominate twenty-first-century science: viewing the mind as a digital computer, and viewing the mind as a statistical machine.

These two alternate frameworks for studying biological thought share the same shortcoming: both are predicated upon principles and mathematics developed for other (simpler) disciplines. Advocates of the “mind is a computer” perspective repurposed old ideas from logic, information theory, and algorithm theory. They tend to view brains as a collection of zeros and ones³. Advocates of the “mind is statistics” perspective repurpose even older ideas from Bayesian statistics and probability theory⁴. These are “top-down” approaches that try to shoehorn Mind to fit pre-existing assumptions established to study phenomenon other than brains and biological thinking. Phenomenon based upon non-purposeful activity.

Grossberg, in contrast, took a “bottom-up” approach that made no assumptions about how Mind might work, instead (like Newton) unriddling the principles as he went along, through experimentation and obsessive calculations.

Grossberg’s quest was aided by a stroke of fortune. Unbeknownst to anyone, including Grossberg himself, the intoxicating puzzle he had stumbled upon in Psych 101 (the serial position effect) was a product of one of the most important and sophisticated neural modules in the human mind, a module essential for music, toolmaking, and language. This module could not be fully understood using any approach other than the Dynamic Mind framework. In effect, it was as if an Astrophysics 101 professor had shared with her freshman class a number of unsolved mysteries, including Venus’s weird backward rotation, Pluto’s oddly shaped moon, and Mercury’s deviant orbit around the sun (a celebrated anomaly that cannot be explained by Newtonian mechanics), and Grossberg not only instinctively recognized that Mercury’s orbit presented the deepest puzzle but also, after studying the puzzle for a year, derived an outline for the theory of relativity.

According to Grossberg’s budding theory, the primacy and recency effects in the serial position curve were “emergent properties” of the holistic dynamics of the mind’s neural networks. This was another radical insight, especially considering that the notion of emergent properties did not yet exist in the mind sciences. Indeed, complexity theory, the intellectual home for emergent properties, didn’t formally exist yet, either.

Though the teenage Grossberg had not been exposed to any neurophysiology, his model had naturally deduced (as a consequence of the mathematics of dynamic systems) several fundamental properties of every biological nervous system, including the existence of individual thinking elements (neurons) that were linked together with directional connections (axons and synapses) that sent signals over these connections if a threshold activation was reached (action potentials). He did not learn that his private model was a good match for physiological reality until he chatted with his premed student friends who were learning neuroscience. This was a revelatory moment for Grossberg. He realized that he had derived the existence and operation of physical mechanisms by mathematically analyzing psychological data.

“I can hardly recapitulate my excitement when I realized this,” Grossberg says. “When it dawned on me that by trying to represent the real-time dynamics of behavior you could derive brain mechanisms, I started reading neurophysiology with a vengeance.”

Beyond any specific model, by the time he graduated from college in 1961, Grossberg had introduced a whole new scientific paradigm for understanding biological thinking: the mind as a marvelously complex dynamic system. Dynamic Mind. He realized that it should be possible to derive a unified dynamics of mind in the same manner that Newton derived a unified dynamics of matter.

In other words, Grossberg believed it was possible to explain phantasmagoric mental phenomena like consciousness, language, and the Self using the same kind of exacting nonlinear mathematics that explained the serial position effect. This was as counterintuitive and heretical as Newton believing you could explain the dynamics of heaven by studying the dynamics of Earth.

There was one big problem, however. The same problem that confounded young Newton. The mathematics supporting a dynamics of mind hadn’t been invented yet.

5.

Just as Isaac Newton had to invent calculus to support his novel intuitions concerning the behavior of physical processes, Grossberg realized that he needed to develop new mathematics to support his intuitions concerning the behavior of mental processes. “Most paradigm shifts in the twentieth century, say in physics, had the math already there to support the new revolutions after new intuitive breakthroughs occurred, including relativity theory and quantum mechanics. Our field is different: we needed new intuitions and new mathematics about complex adaptive systems, because the behavior of these systems was characterized by ‘the three N’s’: non-stationary, non-linear, non-local processes.”

For the next six decades, Grossberg developed new methods, models, and mathematics to support his unorthodox conception of thinking. In 2021, sixty-five years after his groundbreaking insights as a seventeen-year-old hoping to “touch the eternal,” Grossberg published an epochal book entitled Conscious Mind, Resonant Brain. It lays out a comprehensive mathematical framework governing a breathtaking panorama of mental activity encompassing every major form of thinking and every major brain structure, addressing everything from retinal neurons to symbolic communication, the summation of over five hundred articles with more than one hundred collaborators and backed up by behavioral, psychological, neurophysiological, neuroanatomical, biophysical, and biochemical data⁵.

Pick almost any part of the mind, at any level, and Grossberg has put math to it.

The most significant contribution of Grossberg’s unified theory of mind, however, is its solution to one of the most boggling mysteries in science: how consciousness works and why consciousness exists at all. This explanation did not arrive in a sudden moment of revelatory insight but arose gradually over many years through the painstaking accumulation and integration of disparate insights and mathematical processes, as with Principia Mathematica.

And just as Newton showed that an apple falling from a tree followed the same principles as a moon orbiting Jupiter, Grossberg showed that your conscious experience of reading this article obeys the same underlying principles of purpose as a bacterium seeking the light.

.6

You may be wondering why you’ve never heard of the marvelous Dr. Stephen Grossberg if he’s achieved so damn much. Cognitive scientist Margaret Boden, who helped develop “the world’s first academic programme in cognitive science,” explains (2008):

The identification of discoveries (both as new and as valuable) can depend heavily on the discoverer’s rhetorical skills, or lack of them. . . Arguably, one case in point is the early work of the highly creative cognitive scientist Stephen Grossberg. He was perhaps the first to formulate three ideas that are influential today under the names of other people: Hopfield nets, the Marr–Albus model of the cerebellum, and Kohonen self-organizing maps. Grossberg also pioneered many more notions—including back propagation—that are commonly attributed to others, if not actually named after them. . .

The key sense in which his work was ‘‘too far ahead of its time’’ was the unfamiliarity of the mathematics. He was talking about brain and behaviour as a complex dynamical system (as he put it: ‘‘nonlinear, nonlocal, and nonstationary’’), a theoretical approach that didn’t become popular in cognitive science until the late 1980s. But it’s the second point that’s of special interest here: the rhetorical style. His early work was largely unintelligible even to the few psychologists who took the trouble to read it. He combined intellectually demanding (and unfamiliar) mathematics with a host of interdisciplinary details, most of which would be unfamiliar to any individual reader. They were there because he was trying to show the unsuspected theoretical unity behind hugely diverse data. His writing was unusually voluminous too: 500 pages for his first-year graduate report (1964), and many long and richly cross-referenced journal articles. Faced with this challenge from a youngster they’d never heard of, most people gave up before reaching the end, if they could summon up the courage to start reading at all.

Boden underscores the three prime reasons Grossberg remains so unknown and uncited:

1.     His poor communication skills. Most vexing, his inability to provide accessible and accommodating pathways into his work for beginners. Grossberg produced no tutorials, primers, or how-tos.

2.     The onerous intellectual demands of understanding theories, data, and arguments drawn from an astonishingly diverse range of disciplines, including neuroscience, molecular biology, game theory, evolutionary theory, art and aesthetic theory, language and communication, circuits and electrical engineering, biochemistry, brain imaging, anthropology and human evolution, complexity theory, neurophysiology, behavioral science, ethology, psychiatry and mental illness, animal cognition, and much more. To truly grasp and appreciate Grossberg, one must possess at least a passing familiarity with all of the above.

3.     The onerous intellectual demands of difficult and unfamiliar math. In particular, the dynamics of purpose, quite different than the dynamics of (aimless) matter.

But at this point, in the twenty-first century, there’s an even bigger reason why nobody wants to learn and adopt Grossberg’s all-encompassing perspective. A reason perfectly articulated by Tolstoy:

I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives.
   .Leo Tolstoy, What is Art?

In short, it’s a supermind problem.

Most working mindscientists today—neuroscientists, psychologists, psychiatrists, behavioral geneticists, psychophysiologists—have been trained (often for decades) and conducted academic research using assumptions, concepts, methods, intuitions and math different than the Dynamic Mind approach. What this means concretely is that millions of scientists and students around the world have committed years, decades, lifetimes to learning and publishing about the mind from other perspectives—perspectives now tightly sutured together within the global tribe of mindscientists.

What professor wants to admit they’ve been pursuing false trails and red herrings? That the course they taught last semester is mostly misguided? What researcher wants to recast their work in terms of another (contemporary!) researcher, rendering their own work a footnote to the other’s greatness? There’s a reason so many other academics steal Grossberg’s work and pass it off as their own! Peer-reviewed journals, academic conferences, university departments, NSF and NIH and NIMH and ERC funding are mostly dedicated to conceptual and mathematical approaches to the mind other than the Dynamic Mind approach. The administrators at the top of these institutions certainly don’t want to suggest that the billions of dollars they invested over the past few decades was going to fund research on piddling old classical mechanics instead of the smart fast theory of relativity.

Just imagine what it would take for all those folks and institutions to announce, “All right, we were all barking up the wrong tree like Freud and Skinner before us, but now we see Grossberg’s wondrous light!” Especially when Grossberg hasn’t even laid out a clean, accessible, accomodating path to boarding the Dynamic Mind supertrain.

The truly fascinating thing, for me, is that there really isn’t anything comparable to this situation in the history of science. Where a scientist has spent sixty-five years building a scientific and mathematical edifice unifying an entire major field of study, while the rest of the field totally ignored him, despite their own dismal lack of progress. (Just grab a random mindscientist and ask her how consciousness works. . .)

It’s also a reflection of the evolution of science from an endeavor where individual scientists often went off on their own to work out elaborate fundamental theories (like Gregor Mendel, Charles Darwin, Albert Einstein, Isaac Newton, James Maxwell) to a careerist culture, where everyone shares the same basic assumptions and there’s a hundred authors on the same paper and there’s no longer room or possibility for revolutionary frameworks to overthrow vast portions of professional science, because of the grave loss of respect and funding that would occur across the discipline.

The stultifying stagnation of mindscience reaches its zenith in psychiatry, where the bible of scientific psychiatry, the Diagnostic and Statistical Manual of Mental Disorders, is more or less marooned within its commitments to diagnoses established in the 1980s and 90s, because so many massive institutional stakeholders cannot afford significant changes in the diagnoses: health insurance companies, schools and universities, pharmaceutical companies, hospitals and clinics, the military—trillions of dollars of commercial, civil, and government institutions all tightly and expensively stitched up with the existing DSM criteria for mental illnesses. None of them want major changes to our basic understanding of mental illness, resulting in stasis rather than discovery⁶.

Which is why it is highly unlikely that you will see the never-correct and wildly outmoded clinical definition of autism depart from the hundred-year-ignorance embodied in the criteria of DSM 5 299.0 “Autism Spectrum Disorder” anytime soon.

And you can perhaps understand, now, why Isaac Newton and Stephen Grossberg are two of my greatest heroes and role models. Individuals who relentlessly pursued the fundamental nature of reality, uncowed by social norms and supermind politics, unafraid to work alone for decades even when society scorns you or ignores you, because these audacious explorers understood that the pursuit of the deepest truths and greatest mysteries demands a willingness to throw oneself into the furthest reaches of nonconformity.

A natural and comfortable place for anyone endowed with the dark gift.

1. Neuron populations processing a pattern of input signals face a challenge: “If the signals are too small, they can be lost in the noise. If they are too large, they can saturate their respective populations, thereby creating a uniform pattern of excitation across populations and destroying all information about the input pattern. In short, noninteracting cell populations are caught between two unsatisfactory extremes” (Grossberg, 1973). Grossberg terms this challenge the noise-saturation dilemma. He demonstrated that on-center, off-surround shunting dynamics (the membrane equations of neurophysiology) solve the noise-saturation dilemma. “The ubiquitous nature of the noise-saturation dilemma in all cellular tissues clarifies why such on-center off-surround anatomies are found throughout the brain. The cooperative-competitive interactions that preserve cell sensitivity to relative input size also bind these cell activities into functional units” (Grossberg, 2000).

2. As a graduate student at Rockefeller, twenty-five-year-old Stephen Grossberg turned in a five-hundred-page monograph synthesizing the results of his work over the previous eight years, entitled “The Theory of Embedding Fields with Applications to Psychology and Neurophysiology.” His advisers at Rockefeller arranged to have this unusual manuscript mailed out “with a cover letter to 125 of the main psychology and neuroscience labs throughout the world,” including David Hubel, Steve Kuffler, Eric Kandel, and most “major neuroscientists and cognitive scientists in the world at that time.” In Grossberg’s telling, “no one seemed ready to understand it.” His attempts to submit parts of this monograph as ten separate research papers also met with little success: all were rejected. Eventually, when he joined the faculty at MIT, he published numerous papers recapitulating all the work in his Rockefeller monograph.

3. “When we think and perceive, there is a whir of information-processing, but there is also a subjective aspect,” the philosopher David Chalmers asserted, giving voice to the intuitive perspective of countless scholars, scientists, and citizens who readily acknowledge that consciousness must be linked to activity in the brain, but who nevertheless remain hesitant to accept that consciousness is merely the product of activity in the brain—at least, the same kind of activity that controls our breathing and allows us to distinguish black from white. Grossberg:

First, is it fair to ask what kind of ‘event’ occurs in the brain during a conscious experience that is anything more than just a ‘whir of information-processing’? What happens when conscious mental states ‘light up’ and directly appear to the subject? . . . Over and above ‘just’ information processing, our brains sometimes go into a context-sensitive resonant state that can involve multiple brain regions. . . But what is ‘‘information’’? The scientific concept of ‘‘information’’ in the mathematical sense of Information Theory (Shannon, 1948a, 1948b) requires that a set of states exist whose ‘‘information’’ can be computed, and that fixed probabilities exist for transitions between these states. In contrast, the brain is a self-organizing system that continually creates new states through development and learning, and whose probability structure is continually changing along with them.

4. The “Bayesian machine” (or “Bayesian brain”) approach to the study of the mind holds that the brain functions as a probability calculator, coding sensory information as probability distributions and updating these distributions in a manner that is close to mathematically “optimal.”  

Grossberg (2013): “However, the Bayes rule is so general that it can accommodate any system in Nature. This generally makes Bayes a very useful statistical method. However, in order for Bayes concepts to be part of a physical theory, additional computational principles and mechanisms are needed to augment the Bayes rule to distinguish a brain from, say, a hydrogen atom or a hurricane. Because of the generality of the Bayes rule, it does not, in itself, provide heuristics for discovering what these distinguishing physical principles might be.”

Grossberg also asserts, “The mathematical foundations of Deep Learning are contradicted by hundreds of neurobiological experiments. In addition, the algorithm has explained essentially no psychological or neurobiological data. Biological neural models have been developed over the past 40 years that have explained large numbers of psychological and neurobiological experiments, and that have made many successful experimental predictions. These models have also been used in many large-scale applications in engineering and technology, especially those that require autonomous adaptive intelligence.”

5. How does one build mathematical models that link observed behavior to the brain mechanisms that generate them? Grossberg (2017a) notes that linking brain and behavior is “crucial in any mature theory of consciousness, since a theory of consciousness that cannot explain behavioral data has failed to deal with the contents of consciousness, and a theory of consciousness that cannot link behaviors to the brain mechanisms from which they emerge must remain, at best, a metaphor.”

Grossberg refers to the guiding methodology behind his research over the past sixty-five years as the “method of minimal anatomies.” A key idea in this method is that “one cannot derive a theory of an entire brain in one step.” Rather, it demands incremental refinement:

“a kind of design evolution whereby each model embodies a certain set of design principles and mechanisms that the evolutionary process has discovered whereby to cope with a given set of environmental challenges. Then, the model is refined, or unlumped, to embody an even larger set of design principles and mechanisms and thereby expands its explanatory and predictive power. This process of evolutionary unlumping continues unabated, leading to current models that can individually explain psychological, anatomical, neurophysiological, biophysical, and biochemical data about a given faculty of biological intelligence.

A “minimal” model is one for which if any of the model’s mechanisms is removed, then the surviving model can no longer explain a key set of previously explained data. Once a connection is made top-down from behavior to brain by such a minimal model, mathematical and computational analysis discloses what data the minimal model and its variations can and cannot explain. Such an analysis focuses attention upon design principles that the current model does not yet embody. These new design principles and their mechanistic realizations are then consistently incorporated into the model by unlumping it to generate a more realistic model. If the model cannot be refined in this way, then that is strong evidence that the current model contains a serious error and must be discarded.” (Grossberg, 2018)

6. Which I discovered while co-authoring Shrinks: The Untold Story of Psychiatry with the president of the American Psychiatric Association and director of the DSM-5 rollout, Jeffrey Lieberman.