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Computational Neuroscience Physical Theory Lab

Brain function must manifest in the spatiotemporal patterns of neuronal activity.  What’s the secret?  How can we crack the code?  We discuss two properties of brain systems that may be key: the recent finding that neocortical slice cultures show what is called critical homeostasis, and the very old and well known finding that the intact brain shows oscillations.  Critical homeostasis addresses the spatial connectivity-related aspects of these patterns, while oscillations address the temporal patterns.  Disorders of critical homeostasis and abnormal oscillations may result in neurological disorders including epilepsy, mental retardation and movement disorders.

Critical homeostasis

Neural systems that learn are thought to learn by Hebbian association.  If two neurons consistently fire in the right sequence, then the connection between them is strengthened, while if the activity between two neurons is not correlated, then the connection between them weakens.  Computer simulations, however, show that this kind of learning is highly destabilizing.  Favorable connections are made ever stronger, and unfavorable connections are made ever weaker.  Simulated brains that learn by simple Hebbian association evolve inevitably into a state of either escalating overexcitability, or else perpetual silence.  Either state is useless for animals that learn.

Real brains maintain homeostasis of neuronal firing and connectivity in some suitable range [1-5].  Furthermore, experiments in neocortical slice cultures show that connectivity is maintained at a level that actually maximizes the variety of possible spatial patterns of activation, and that minimizes loss of information transmitted through such a system [1, 2].  This special level of connectivity is referred to as “critical” connectivity.  In addition, the same spatial patterns of firing can spontaneously recur over and over again, hours apart [6, 7].  That is, information can be stored in these spatial patterns of activity. 

If each spatial pattern of activation is recognized as a fundamental unit of information, such behavior is highly desirable for a system that is supposed to learn.  To borrow language from the physical sciences, these spatial patterns of activation are referred to as “neuronal avalanches” [1], and the cultures are said to show “critical behavior” [8].  One also refers to such systems as exhibiting “self-organizing criticality,” meaning that they are capable of maintaining critical connectivity spontaneously, without external intervention.

What consequences does critical homeostasis have on brain dynamics?  And what can go wrong?

We constructed a simple computational model consisting of a certain number of nodes, each node representing a certain number of neurons [910].  Every node can fire spontaneously, with no input from other nodes, with a probability S, with a different value for each node.  Every node can also cause any other node to fire, with some probability P, different for every pair.  At any given time, the S’s and P’s are scaled up or down, depending if the firing rate and activity of a given node is too high or too low.  We then ran thousands of simulations with different values of the scaling constants, for tens to hundreds of millions of timesteps each [10]. 

To our surprise, homeostasis of activity and connectivity imposes strong constraints on the system.  One constraint is that S must be greater than zero; that is, the brain is spontaneously active.  Most of the activity is due to stimulated (connectivity-related) firing, but S cannot be zero or the system is unstable.  This is why the brain is spontaneously active, even at rest and even in sleep.

Dynamical stability also requires that the scaling of P be faster than scaling of S.  Further, if S falls below its steady-state value, then P rises in compensation, and vice versa. A simulation of a seizure showed that a prolonged seizure drives S to very low values, and hence connectivity rises in the post-seizural state and remains elevated for hours to days.  Connectivity does not recover to baseline until S gradually declines to its own pre-seizural value. 

That the post-seizural state is overconnected for a long period of time means that it can “burn” into memory hypersynchronous, hence epileptogenic states. That is, the supercritical state is epileptogenic.  Furthermore, increasing spontaneous activity, for instance by electrical stimulation, may suppress these hypersynchronous states, and thereby prevent epileptogenesis.  That is, counterintuitively, increasing spontaneous activity is anti-epileptogenic.

The idea of overconnectivity causing disease states may apply to Parkinson disease as well.  The standard theory of this disorder is that the globus pallidus interna (GPi) is overactive in this disease which then inhibits thalamic neurons which then results in clinical hypokinesis [11-14].  This theory motivated resection of GPi and, less drastically, implantation of a deep brain stimulator (DBS) into the subthalamic nucleus (STN).  Since STN provides excitatory input into GPi, the idea was that lesioning STN should decrease output from GPi.  DBS has been very helpful for many symptoms of Parkinson disease.  However, recordings from GPi show that DBS stimulation of STN increases activity in GPi, rather than decreasing it [15, 16].  If DBS stimulation of STN results in increased activity in GPi, then why does DBS help alleviate Parkinsonian symptoms?  This question is unanswered. 

Our conjecture is as follows.  The loss of dopamine input in Parkinson disease causes a decrease in the spontaneous activity of the basal ganglionic circuit.  In compensation, the connectivity within the circuit increases above critical levels, thus inducing hypersynchronous activity in GPi and possibly elsewhere.  DBS stimulation of STN increases the activity of GPi neurons, which relieves some of the drive towards overconnectivity, making connectivity at least less supercritical.  Dysfunction of the basal ganglia circuit in Parkinson disease, by our hypothesis, is thus due to connectivity being supercritical.  Partial relief of overconnectivity therefore alleviates symptoms.  An MPTP model of Parkinson disease has indeed shown hypersynchronous behavior in GPi [17]. 

Oscillations

The study of the basal ganglionic circuit is intriguing for another reason.  In vivo recordings show that there are discrete oscillations that exist simultaneously, which are sharply defined but which only last for about 200 msecs at a time, to yield to another entire set of discrete oscillators.  Transitions between sets of oscillations appear to occur every 200 msecs or so [Montgomery, unpublished].  Similar oscillator set shifts have been described in intracranial neocortical recordings by Walter Freeman [18].  What is the meaning of these oscillations, and why do they suddenly shift all at the same time? 

At the descriptive level, oscillator set shifts can be described in terms of the dynamics of a theoretical particle on an effective potential energy surface [19].  In this approach, one regards the output of each electrode or microelectrode as the spatial coordinate of a theoretical particle.  Specifying all the coordinates of this particle corresponds to specifying the activation at each electrode.  If there are N electrodes, one can show that the dynamics of this particle is that of a particle moving on an N-dimensional potential energy surface.  Oscillatory behavior corresponds to the particle being transiently trapped in a potential energy well, while sudden shifts in oscillator frequencies correspond to escape from one potential energy well into another.  Every potential energy well has its own set of oscillators. 

Using this conceptualization, one can identify the “location” of all significant potential energy wells.  Each location corresponds to a particular spatial pattern of activation, i.e., to a basic unit of information.  As one follows the theoretical particle in time, it will jump from one well to another approximately every 200 msecs.  Specifying the trajectory of the theoretical particle on this potential energy surface (that is, which well is visited in which sequence) specifies the spatiotemporal pattern of a thought. 

Thus it may be of interest to explore the topography of this brain potential energy surface, and to quantify the transition rates of a theoretical particle on this surface as it hops between energy wells.  One might hypothesize that abnormally low transition rates result in hypokinetic movement disorders, while abnormally high transition rates result in hyperkinetic movement disorders.

In vivo structural connectivity

A final area of interest is that of in vivo structural connectivity.  It is possible using magnetic resonance imaging to estimate a quantity called the diffusion tensor, which is a 3 by 3 matrix defined for every voxel that gives the preferred directions of water diffusion in that voxel [20].  Free water tends to flow parallel to white matter tracts, not across them.  By choosing a seed voxel and projecting an imaginary streamline from that voxel in the direction of greatest preferred water diffusion, one can map out a line that is suggestive of a single strand of a nearby white matter tract.  To map out realistic white matter tracts, one has to repeat the process with multiple other seed voxels, then combine the resulting single strands.  Of course, real white matter tracts are not single fibers, but branch repeatedly.   In addition, nonbranching single strand white matter tractography is sensitive to noise, because a single noisy voxel in the pathway of a single strand will throw off the rest of that strand. 

We have a developed a method of diffusion tractography that allows multiple branching.  We first define a 3 by 3 matrix linking every pair of voxels (the “W-matrix”).  This matrix is a product of three factors, accounting for how strongly a preferred direction is specified in each voxel (referred to as the fractional anisotropy), how closely this preferred direction in one voxel is lined up with that of the other voxel, and how far apart the two voxels are. The W-matrix is then used as a transition probability in a computer simulation.  First we choose one or more “seed voxels”.  We make these seed voxels “fire” at a high rate, which then causes other voxels to fire.  The simulation is continued until steady state is reached.  Those voxels that fire at the highest rates are those most strongly connected by white matter tracts to the seed voxels. 

Applications of this project will be towards the diagnosis of disorders of connectivity such as autism [21, 22], as well as to white matter disorders such as multiple sclerosis and Alzheimer’s disease.  Furthermore, of relevance to the projects on epilepsy and Parkinson disease, we hope also to look at hippocampal and basal ganglionic structural connectivity, in health and in disease.