How to Mathematically Measure Consciousness

How can we measure the phenomenon of human consciousness?

A month ago I was being introduced for the public presentation of my neuroscience PhD defense. My mentor, Giulio Tononi, noted that it is remarkable now that there are places, including his own lab, where a young scientist can now pursue a career studying consciousness.

After all, for nearly a century, scientific work on consciousness was considered too strange, too weird, too philosophical, to be “real science.” It used to be that to study consciousness as a scientist you needed the level of immunity afforded not only by tenure, but also a Nobel Prize.

Indeed, the contemporary field was inaugurated in 1990 by Francis Crick (Nobel-Prize-winning co-discoverer of DNA) and Christof Koch. Together, they published the paper “Toward a Neurobiological Theory of Consciousness,” which argued that the time was ripe to scientifically investigate consciousness. In a genius move of diplomacy, they redirected the goal of the investigation to be merely a description of the neural correlates of consciousness (or NCC). Leave aside philosophical positions, they advocated, and concentrated on finding what neural events correlate to consciousness.

After 25 years it’s finally verging on mainstream. When I attended the annual conference in 2010 for the Association for the Scientific Study of Consciousness (the ASSC) it had barely 300 attendants. Last July it was held in Paris and it had doubled in size to over 600 people, so that at the end of the conference there were enough drunken scientists to take over a significant portion of a bank of the Seine and confuse the tourists with talk of consciousness as the sun set.

The fruits of the field are plenty. For instance, there is now an agreed-upon definition of consciousness: it’s the experiential world of sensations and thoughts you wake up to in the morning, and it’s what disappears when you fall into a deep dreamless sleep. That is, “consciousness” as used in the literature now does not mean meta-cognition or self-consciousness or language use or rationality or any of the things it’s been conflated with in the past.

Like many fields, it begins with this working definition and then becomes more precise as it goes along. This communal reference is just one instance of the field’s greatest achievement, which is the development of a useful shared argot between scientists and philosophers. Scientists who work on consciousness now know what “qualia” means (the quality of experience, such as the redness of red) and can often recite a host of philosophical thought experiments, and philosophers of mind now know the difference between fMRI and EEG and can discuss details of experimental design.

However, there has been less progress in finding a single neural correlate for consciousness. This is not just because of the coarseness of our still-primitive neuroimaging techniques, but also the challenge of interpreting neural activity.

One problem is that consciousness may not be localized to any particular area, and much of the contemporary methodological tools of neuroscience have an implicit design for studying localization. Additionally, no matter how many theoretical sandbags you stack up to try to protect your supposedly philosophically-neutral endeavor, theoretical and foundational questions about consciousness always seem to seep back in through the cracks, reminding us that there is an ocean out there and we are still on the shore.

For instance, the big issue at the latest ASSC conference was how to measure the neural correlates of consciousness separately from the neural correlates of reportability. Until now, researchers have generally relied on a participant’s immediate introspection to track if the participant was aware of a change in his or her perception. If the Necker cube is facing toward you, click the right mouse button. If it flips to be facing away from you, click the left. The problem is that every time we probe consciousness we are forcing a report from someone, so then how can we tell the difference between report and consciousness?

In quantum physics the very act of observation changes the phenomenon that is observed—in consciousness research, the very act of reporting the phenomenon may change its neural signature. Or, farther out along the same track of thought, consider that we are aware of much more than we can report at any one time. This fact, given its academic name by the philosopher Ned Block, is referred to as “phenomenological consciousness overflows consciousness.” So while we are constantly immersed in a lush world of sensations that contain information about locations, color, meaning, motion, and so on down a very long list, we can only briefly pick out and summarize small parts of it at any one time. Our cups runneth over.

It’s often these types of abstract knots that cause eliminativists (perhaps the most notable being the philosopher Dan Dennett) to throw up their hands and declare that consciousness simply must be an illusion. But on further inspection most eliminativist claims end up eating their own tails: after all, an illusion is when something appears different than the reality, but in the domain of consciousness the appearance is the reality. Without going to such extremes and embracing such a counter-intuitive conclusion that shuts down the research entirely, there’s been the rise of a new scientific approach to consciousness.

Bees, octopuses, the artificial neural networks driving around cars now—these are all systems, some biological, some engineered, which we don’t actually know are conscious or not. Is there anything it’s like to be a self-driving car? Imagine a “conscious-o-meter” that you could point at each of those things and it would tell you both the level and content of consciousness.

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At the cutting edge of consciousness research people are asking: how would such a hypothetical conscious-o-meter make its decision? This is the search for a mathematical measure of consciousness (or MMC). The field is built on the hope that, like the rest of nature, the book of consciousness is written in the language of mathematics.

For example, consider that in the search for the NCC, the original hypothesis for the correlates was cortical neurons oscillating around the 40-70 Hertz frequency (what neuroscientists call the “gamma band”). It was given by Francis Crick and Christof Koch in their 1990 paper and purposefully acted as a simplistic but useful starting point. The analogous hypothesis in the search for a MMC is that if a system passes a certain level of complexity it becomes conscious. That’s almost certainly not true (just like the 40-70 Hz oscillations almost certainly isn’t true) but it’s a definable starting point. The proposed measures grow in sophistication from there, drawing from computational complexity theory, information theory, and the latest in causal analysis.

What’s so new about this latest line of research is its focus on formalism, which is why physicists, like Max Tegmark, and computer scientists, like Scott Aaronson, are participating in the field. Formalism also helps clear up the hand-wavy notions of “information” or “processing” or “representation” that get thrown around when talking about consciousness, and often fool people into thinking that they have some obvious answer to its mystery.

Ultimately, the search for a MMC proceeds by asking: what properties should a mathematical measure of consciousness actually capture? A good method is to start with the properties of our own experience and note that it has certain properties, such as always being unified (you are one person experiencing one subjective world). These are good properties for a MMC to be sensitive to.

It’s this reasoning that led to the development of what is arguably the leading measure of the field, called Integrated Information Theory (or IIT). Originally conceived by neuroscientist Giulio Tononi around the turn of the millennium, it has since been updated several times to take into account theoretical and mathematical advances. During my time as a PhD I was part of the team of theorists, led by Tononi, that developed its latest incarnation. In its updated form, IIT says that it’s not actually the information (as traditionally conceived) in a system that matters, but rather whether the system’s causal structure reflects essential properties of consciousness.

For example, does the system form a causal structure that is highly integrated? This could account for the fact that consciousness is always a unified experience. Do the parts of the system all have distinct causal roles, and act in combination to produce effects? This might account for how our experiences, while being unified, are at the same time composed of all different sorts of sensations and thoughts bound together.

Note that we’re not just talking about brains here. Rather, the mathematical correlates of consciousness are purposefully substrate-neutral. There’s no magical fairy dust in neurons that grants them consciousness, and, since many of the proposed measures could be just as easily be applied to systems that are not the brain (such as computers or communication networks) it suggests that consciousness may not be confined solely to the biological domain.

Historically, consciousness has been a subject of recurrent debate, first rising explicitly as a subject of scientific study under Wilhelm Wundt and his students in the late 1800s. Originally, it was thought of as a major part of the field of psychology that was being founded at the same time. Wundt and his followers relied heavily on their own and participants’ introspection to try to ask questions like: what is the simplest sensation? It was precisely this reliance on report that behaviorism critiqued so viciously, so devastatingly, that the entire approach was forgotten and the eliminativists ruled all through the next century.

At the latest ASSC I saw the same historical line of argument rear its head like an unkillable gorgon, and even in the midst of scientific posters and Paris I became afraid that history will repeat itself, that consciousness research will again become considered too problematic, report too small a window, introspection too untrustworthy, the methodological issues too open to attack.

But this latest interest in formalism and the MCC avoids the problematic emphasis on empirically probing consciousness. Rather it takes consciousness as a natural phenomenon begging to be described, a property or process inherent to certain types of complex systems. Perhaps by having something that Wundt and his followers didn’t have we can avoid the same fate.

Of course, maybe this is always the view from the top of the wheel, but to me it sure feels like it’s different this time.