To describe the whole brain, we need to introduce state variables for all of its neurons — and there are about 100 billions of them — and to form the joint state space, because the neurons are connected. So the dimension of the state space of the brain is really large, much larger than 3.
So the second step in constructing a model is to specify the rules which govern the behavior of the system. And in our terms these are the rules which govern the motion of the state point in the state space. We do it as follows: At every point in the state space we put an arrow which has a certain length and a certain direction. This arrow is called the velocity vector. The full collection of such arrows is called the velocity vector field. It works as follows: Imagine that our state point is represented as a football moving in the state space. And imagine that at every point in the state space we put a footballer who does two things: He accepts the ball from where it arrives and kicks it in a given direction with a given speed. In between the two kicks, the football travels for just a tiny amount of time before it is caught by the next footballer. And the next footballer kicks it in a different direction with a different speed. That is why the football moves.
So this metaphor explains why our field of arrows is called the velocity vector field: it is because speed with direction is called ‘velocity’, and in mathematics all quantities with direction are called ‘vectors’. So we know that if we consider a car moving along a curved village road, the velocity of the car tells us how quickly and in what direction the car moves in the physical 3D space. But here the velocity vectors tell us how quickly and in what direction the state point moves in the non-physical state space.
Interestingly, when all these arrows are permanently fixed in time, do not change, they still prompt the system to evolve, to behave, and they fully determine all properties of this behavior. So the velocity vector field of the device plays the role of the ruling force which dictates the system what to do in every possible situation.
You can see that this object is much more sophisticated than a straight line or a surface. And, in fact, it was introduced by Henri Poincaré, a mathematician — largely thanks to him — especially to handle complex systems. The actual model has this form, and it is called a dynamical system. These equations mean that, at every point in the state space, we have a velocity vector whose components are rates of change of individual variables, or velocities. The values of these components depend on the location in the space, as we say in mathematics, they are functions of this location.
How do we know what these values are? Well, in this Universe all devices obey the same laws of physics and chemistry, and that includes living systems and the brain. In brain models, these components are simply the combinations of physics and chemistry laws which we apply to various parts of the brain with account of the brain’s architecture, such as the connectivity map. So essentially, the velocity field of a device is simply a mathematical representation of the physical structure of this device. Living systems have been modelled as dynamical systems, and perhaps the most famous neuron model in this form is due to Hodgkin and Huxley and is now over 60 years old.
More recently, there were several attempts to model the whole brain as a dynamical system. So, ultimately, to build the model of the brain means to determine the brain’s velocity field. It is important to appreciate that when we observe the brain activity in an experiment during finite time, we only observe just one trajectory in the state space of the brain, which corresponds to only one behavioral pattern of the brain. But the velocity field holds information about all possible behavioral patterns the given brain can exhibit in principle.
To resolve the mind-body problem, I propose to identify the mind with the velocity field of the brain. Also, to the question: How is the brain connected to behavior?, my answer is: through the velocity vector field which the brain creates. Let me explain how. Go back to the list of features of the mind, and now — shuffle them a bit. And now compare these features against the features of the velocity field of the brain. Is our velocity field a field? Yes. Is it called into being by the brain? Yes, there is no velocity field of the brain without the brain. Does it emerge from interactions between the brain components? Are its properties determined by the physical properties of the brain? Yes and yes, most certainly.
Does it govern our behavior? Let us see. Suppose your body moves. What makes it move? Muscle contraction. What makes muscles contract? The appropriate coordinated firing of special neurons, called motor-neurons, which connect the brain with the muscles.
What makes motor-neurons fire appropriately? The firings of the neurons in the brain to which they are connected. And these are governed by the velocity field of the brain. So, ultimately, the velocity field of the brain governs the behavior of our body. Is it distinct from the brain? Yes, if only because, unlike the brain, it exists in the non-physical state space. Does it coincide with the brain? In a certain sense, yes, because the velocity field of the brain is only a mathematical representation of the brain’s physical structure. It is a bit like saying that what you see on this slide is a straight line, but all you see here is a collection of ink particles deposited on paper.