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Wednesday, May 5, 2010

Theory of Functional Brain Disorders- for Physicians

We do not have a formal description of Le Chatelier's Principle for non equilibrium processes. The concept of equilibrium can be formally specified for gases and liquid phase solutions. The concept has also been used loosely as an analogy, in describing receptor dynamics (sensitization, resistance, and metabolic control of receptor number and maybe distribution).

When the principle is used for receptors, it loses the mathematical underpinnings (which requires permutations and combinations that underlie the concept of randomization). That is, once we have begun to use the principle for describing receptor equilibrium, we have begun to use analogy, not thermodynamics. Still, when we just look at how receptors act, the use of equilibrium as metaphor is a very useful simplifying concept.

So, we have gone a little down a pleasant road in which we realize that equilibrium processes can be invoked in non equilibrium processes, but we have no formal description of this, because we have no mathematics for it. From a thermodynamics point of view, the improbable things that populate our world, including life and the especially thermodynamically improbable objects of the study of neurobiology, are thought of as open systems that require a sustained energy gradient in order to maintain their structure against ambient temperature fluctuations. Jayne's MaxEnt principle suggests that the appearance of stable objects in a non equilibrium system solve some optimization problem for energy dissipation. But, in general, the mathematical perspective is that non equilibrium processes are uninteresting- because the boundary conditions aren't specified, the problem for instance of what the brain is doing in terms of structure and energy dissipation is that this happens to have been a symmetry break somewhere in the environment, and that leaves neurobiologists with cleaning up incidental details.

In clinical medicine, we sometimes go a little farther down the pleasant road, when we say such things as "nerves stabilize nerves". There is Cannon's idea of homeostasis, which invokes the metaphor of equilibrium. In neurology and psychiatry, we have the concept of disinhibition- by which they mean roughly, I think, that when we consider the nervous system as distinct populations, and a lesion occurs in one, changes occur in the rest of the nervous system that reflect the inhibitory effect of one population of nerves on another. Processes in which the equilibrium metaphor is used in clinical medicine include the hyperreflexia of lower segmental reflexes in upper motor lesions, hypothalamic and pituitary activation during the cortical inhibition of sleep, perhaps the "positive and negative manifestations" of schizophrenia, and perhaps the role of the failure of protective inhibition in the generation of excitatory damage.

However, the metaphor of equilibrium is further removed from the mathematical concept of equilibrium as a practical matter, because we lose a clear notion of symmetry with this extension. For example, a spinal transection "causes disinhibition of lower segmental reflexes"- but does a spinal transection cause disinhibition of cortical processes? At least with the extension of the metaphor to receptors, we continued to grasp the conceptual tool of symmetry. With the concept of disinhibition, we kind of go quiet on this. So, does the "disinhibition" of lower segmental reflexes after stability is reached in spinal transection predict a general cortical hyperreflexia? Interestingly, Brown-Sequard said it did, and thought he produced epilepsy in dogs by spinal transection ("Brown-Sequard Epilepsy").

The use of symmetry arguments at the level of the homeostatic organism is complicated by the fact that the nervous system in vertebrates has only bilateral symmetry. Characterizing the effect on lesions in the antero-posterior axis in terms of equilibrium poses the further difficulty of extending the metaphor of equilibrium, because we have no mathematics to describe the act of dividing a nonequilibrium system in half. That is, there are chunks in there.

So, there is a mathematical disconnect between the thermodynamics of computation in the physics literature, and characterizing neurobiology (and biological processes in general) as computation. A quick summary: Landauer's one molecule model of the bit invokes inserting a division in the phase space of a molecule; if the one bit of knowledge of which side this molecule is on is used to insert a piston on the division, the division of phase space can be used as a Carnot heat engine. The cost of resetting the memory register after expelling entropy to the environment is kTln2. Bennett later made precise the concept of reversible computation. So, the field of computational thermodynamics grew up on this firm mathematical foundation, whereas neurobiology grew up on the foundation of biological research on cells, genes, and receptors.

In biology today, there is an emphasis on receptors, membranes, and genes, and a relative neglect of biological processes in terms of dissipation of energy gradients and metabolism. Brain circuits are the province of poorly funded device companies; brain receptors are the province of well-funded pharmaceutical companies.

In neurobiology, the McCulloch and Pitts neurons, and later Minsky's "perceptron" machines, showed that one could construct logical sentences from neurons with Boolean logic characteristics. The formal descriptions of neural nets showed that simple permutations and combinations of such logical simples can produce any computation. The work of Hubel and Weisel on the processing of vision in the cat is physiological evidence that the brain works at least in part as a logic engine.

So, historically, after it was shown that a neural net functions as a computer, the popular but mistaken assumption was that a neural net functions only as a computer. The computer brain is great at chopping things up- but it isn't good at putting things back together. That is, we may have a cell in the brain that responds only to the appearance of your grandmother, but how do we find it, relate it, how do we rotate images mentally for example? When one chooses to walk through one of two doors, how does one funnel all this analysis into a decision? This has been called the "binding problem". It is a problem in the sense that if you are straining the McCulloch/Pitts computer metaphor beyond it's applicability, you have caused your own puzzle: the binding problem is a disrupting issue for the brain computer metaphor. The binding problem is a philosophical gap precipitated by excluding non analytic functions from thermodynamics, rather than by seeing classical thermodynamics as a subset of non analytic processes.

So, if you are a neurobiologist and you want to talk more precisely about such things as inhibiting, exciting, and dividing, inside of a population of cells "at equilibrium", one needs to know what one can say mathematically about non analytic set mappings. Between any two sets of elements, one can construct set relations that preserve the notion of a function. And as long as you are standing on the ground of set mappings that meet the criteria of a function, this allows one to bring in the mathematical tools of algebra, calculus and differential equations. You see these tools used frequently in neurobiology, for example in the use of differential equations to model neuron behavior.


I'm going to try to make clear a more general account of equilibrium below.


Most processes in nature involve set mappings that are not functions. These are mappings in which information is lost, or gained. That is, in the universe of possible mappings between any two groups of elements, there is a subset of mappings that are functions, and there is another group in which mappings are few to many, many to many, or many to few.

To take a biological example, one can say that information from ambient light rays are lost when they enter the eye (from positional orientation, pupillary constriction, and variability of the ability of different points on the retina to respond to different stimuli such as color or movement). The transformation from light rays to neural reception in the retina is an information-losing mapping, i.e. a "many-to-few" mapping.

The processing that occurs between the retina and broadcasting into the brain is not a function either. This mapping is few to many, as there are many more cells in the brain that there are in the retina. It is an information gaining mapping, the inverse of an information losing mapping. (To construct this notion of information gain precisely, one would use Chaitin's definition of algorithmic complexity, and show that it would take a longer string to specify the brain state containing the projections than it would to specify the state of the retina).

So, we now have a topology of brain mappings that is free of of the requirement that the mappings fulfill the definition of a function. And, we are leaving behind the tools of analytical mathematics. Instead of function analysis, we now base set mappings directly between any number of systems at homeostasis, or when a system at homeostasis is divided (lesioned or transected).

I call this approach Confocal Set Theory (maps with lenses), because it looks at the information-losing and information-gaining transformations that occur when set mappings pass through a convergent or divergent lens. A convergent mapping occurs onto the retina, and a divergent mapping occurs into the brain in central projections. Confocal Set Mappings are non-function relations between sets in which information is compressed or expanded.

Looking at neurological circuits as information compression or expansion set mappings allows us to formally extend Le Chatelier's principle to equilibrium processes in non equilibrium systems at homeostasis. The homeostasis is defined in terms of the persistence of the whole organism over time. Whole organism homeostasis replaces random processes as the starting point for discussing division of a phase space.


I'm going to apply CST to the pupil below, and make a case that brain circuits are pupil-like transformations.


When the pupil is maximally dilated, mappings from points in the visual fields to points on the retina are many-to-many. This holographic mapping has maximum information (maximal information in Chaitin's sense that a random sequence of numbers has maximal information; here each retinal cell sees everything) and minimal focusing. There is no spatial organization transmitted through the maximally dilated pupil, and to extract this information requires further McCullock/Pitts processing. Now you have a problem explaining mental rotation of images, you have a memory access problem, you have a binding problem.

When the pupil is maximally contracted, mappings from points in the visual fields to points on the retina are one-to-one. The image is focused as a consequence of the information compression mapping by the eye and pupil.

When the pupil is at an intermediary position, some non-analytical preservation of spatial relations occurs to an intermediary extent.

The pupil is an example of CST mappings. There are two characteristics of convergent or divergent information mappings. One is that the compression or expansion work done on the information stream is a type of spatial relation computation that is entirely distinct from the McCulloch-Pitts logic engine computer. The second characteristic of convergent or divergent information mappings is that the spatial relations are inverted.

So, we now have another metaphor for neurobiology: we have the brain as a digital computer, and we have the brain as an analog camera obscura.


I'm going to make the case that pupil-like processing occurs throughout the CNS


When light rays are focused on the retina, there is a one-to-one mapping preservation that fulfills the definition of a function. However, information compression and expansion mappings in general continue to exhibit the same spatial-preserving behavior, even though the mappings no longer meet the mathematical definition of function.

So, between populations of excitable cells- the retina and the LGN, the LGN and the visual cortex, the motor cortex and lower motor neuron, between any two sets of excitable cells in the nervous system, there is a white matter tract. The white matter tract is a constriction that performs a pupil-like function in the nervous system.

The concept that white matter tracts are performing pupil-like functions of information compression and expansion in the brain suggests a way of understanding the somatotopic preservation and inversion of the homunculi in the motor and sensory cortex, and the crossings (decussations) in bilaterally symmetric nervous systems. As far as I know, there is no other framework to explain evolutionary development of of decussations and somatotopic inversion in spite of the increased length of the tracts incurred by the crossing and what advantage this metabolic cost has (the metabolic cost comes from the fact that crossing tracts are longer).

The metabolic control of white matter tracts occurs through the same autonomic regulation as that of the pupil; blood flow to white matter tracts is regulated by autonomic functions. This autonomic regulation allows pupil-like adjustment of the aperture of the white matter tract. There is a metabolic cost to maintaining the far branches of an arteriolar tree, and vascular flow probably regulates how far the activity of these branches extend into the white matter tract, and thus regulates the size of the white matter aperture.

Brain metabolism controls the trade-off between the brain as a camera obscura and the brain as a computer.


Brain metabolism as capacitors in parallel

When we talk about processes extending between neuron cell bodies, we describe them as excitatory or inhibitory, in analogy with a binary logic gate in computers. One consequence of restricting the view of the brain to functions is that it becomes impossible to think broadly about the effect of a disturbance in equilibrium on a brain at homeostasis. What is the effect on the overall relationship between two populations of cells when you inhibit an inhibitor? What about the effect on the excitatory input to the inhibitor? Paying too much attention to this tangle of excitation and inhibition is the neurobiologist's version of the binding problem.

When we allow mappings that are not functions, we can allow ourselves to simply describe the large-scale circuit effects. The levels of chemistry, receptors, and neurons, turn into "black boxes" and we are in the realm of the effects of populations on populations, mediated by pupil functions of tracts.

So, we have a way of thinking about populations of neurons in the brain in terms of equilibrium reactions- what the cortex does to the HPA axis, what a ganglion does to its projections, in general terms. Le Chaterier's principle generalizes to: Nerves Stabilize Nerves. Populations of nerves stabilize populations of nerves. And the heck with whether they are inhibitory or excitatory, that's in the black box.

So how does this apply to a disturbance of homeostasis of these populations?

I think of these populations- either a single neuron or a population of neurons rendered distinct from another population with which they are in homeostasis through a white matter tract- as being essentially capacitors in parallel. Their energy gradient comes from blood flow.

The sink for this population of capacitors is the energy cost of the radius of the tree of their projections. Only at peak energy use in a flight response can the full metabolic potential of the extent of the tree be used. A sympathetic discharge such as occurs in a flight response leads to dilation of the pupil (through contraction of the pupillodilator muscle) and constriction of the white matter tracts (through diminished brain blood flow). The retina receives maximal information, while parallel processing in the brain is restricted through reduced blood flow. During blood flow restriction in the brain from sympathetic discharge, digital parallel processing is reduced in favor of analog somatotopic projection.

Essentially, the role of the autonomic nervous system in the brain is to regulate the amount of parallel processing, which it does through regulating blood flow in the white matter tracts.


How does an external stimulator affect populations of neurons?


One effect that is known is that chromophores and mitochondria cycle ADP to ATP, that this increases the blood flow to the area, and that a reflex reduction in flow occurs downstream which is mediated through the trigeminovascular system that follows the projection path. The specificity of reduction in the trigeminal evoked potential (rather than in the median evoked potential when the face is stimulated) suggests that the window of view the brain has of the trigeminal pain input is controlled by the autonomic modulation of the sphenopalatine ganglia.


So, what causes migraine?


Migraine is not a brain disease in my view. Migraine is a systemic disease involving all organ systems. In my view, for the broad population of headache suffers, migraine is a metabolic disease, in which intermediary metabolism is disturbing mitochondrial function, and in which tissues that are most metabolically active are affected first. So, it seems to me that the basic problem is a brown-out. We see the cortical spreading depression because that's where the fMRI light is.

It's interesting to me to consider the systemic metabolic effect (endocrine disruption, whatever is causing us to become fat and menstruate early) as primary; cortical spreading depression as secondary; and activation of events in the hypothalamus and brainstem and peripheral nerves (trigeminal, optic, auditory) as equilibrium reactions.