Summary of Results

I.  CA1 Model Development

The software development effort centered around constructing the framework for a realistic model of rat CA1. This ground has been laid as an object oriented MATLAB/NEURON model of pyramidal cells. The move to object orientation is meant to provide for ease in expansion, specifically the addition of interneuron models and synaptic interconnectivity. The model also allows for measurement of the extracellular potential at any number of points, enabling the study of layer polarization for dynamic assemblies. The dynamics of these dipole layers are believed to be fundamental to the LFP characteristics. This model aims to enable study of this dynamic in an effort to bridge the gap between microscopic neural features and mesoscopic field theories. 

Neuron d151

6 Probe Recording Sites

Cell Assembly
1681 Neurons 280µm×280µm

Population Spike Dipole,
3ms Asynchrony Induced Low-Pass Filtering

θ-Entrained Population Spike

Power Spectral Density

II.  Analysis of Spectral Response:  Low-Pass Filter

This model, along with an analytic approximation, were used to study the effect of population spikes on the spectral characteristics of the LFP. Two primary parameters affect the spectral response: synchronicity, and neuronal density. The two are intrinsically coupled, hinting at a more suiting scaling parameterization...The spectral effects produced are due to an  increase in temporal spike density.   As one increases the neuronal density, one brings another shell of neurons into contributing range.    This essentially has the same metric effect as increasing synchronicity.  That is to say, if one increases the density, one can relax the synchronicity to achieve the same qualitative response.

In constructing a stochastic representation of the power spectrum, one must properly calculate the first order moment and second moment (covariance) to describe a model.  The second moment requires a double sum over the time differences of pairwise selections drawn from the entire neuron population.  This can be seen in e.g. the Volterra expansion of a signal:

$$ f(t) = f$$₀$$ + ∫ K(τ) s(t-τ) dτ + ∫ K2(τ1,τ2 ) s(t-τ1) s(t-τ2) dτ1 dτ2 + ...

However, it is easy to realize that the first order correlation corresponds to the limit of infinite temporal spike density.   Therefore, as we increase temporal spike density, the second moment vanishes and one recovers a form proportional to the first order correlation, which is just a convolution integral over the PDF.   This term contains the low frequency information, and represents the limits of any low pass filtering that arises. 

Imagine beginning with an infinite neuronal density and total synchrony, and slowly introducing jitter.   Each term is then a multiple of the same integral and one must only calculate the first order correlation up to a normalization factor.  Now slowly decrease the neuronal density.  The system becomes quantized, destroying our symmetry killing the second moment.  Power is then transferred from our continuum approximation into the covariance term. 

Now, the action potential has a finite width and distribution, serving as a sort of uncertainty relation that defines how quickly our continuum approximation breaks down into a noticeably quantized system of neurons.  A high degree of synchrony in a sparse collection of neurons approaches the continuum limit just as a relatively high asynchrony in a dense collection of neurons approaches the continuum limit. 

If the width of the spike is on the order of a few milliseconds, it stands to reason that asynchrony on the order of a few milliseconds will begin to reveal transition effects.  The strength of these effects depends on the neuronal density.  For high packing densities, as in stratum pyramidale, the system quickly approaches the continuum limit; the power spectrum is sufficiently described by only the first order correlation.  The validity of this simplification is demonstrated through Monte Carlo experiments, and then the continuum model is used to analytically describe the effect of synchronicity/density on the resulting power spectral density.

 Analytic Spectral Shift
(continuum model)

Monte-Carlo

Spectral Shift and Low-Pass Behavior

Analytic Prediction of Low Frequency Behavior

III.  Neural Field Theory and Cellular Automata

Finally, various neural field theories were examined in an effort to assess the compatibility of a theory with the known dynamics and correlations in CA1. Various efforts exist in the literature to describe the empirical power spectrum of EEG, but lack a corresponding spatial description. Conversely, spatially dynamic field theories exist, but lack a good corresponding frequency description.  Furthermore, existing descriptions typically involve drastic simplifying assumptions in order to make the model calculable, and typically offer little insight into the neurophysics at ground-level.

Computability is then a real concern when constructing a field theory that can be realistically implemented in modeling. In general we find field theories that have no known general solutions, and therefore only specific cases that lend themselves to solutions are sampled.  It is typically dubious that even a numerical solution can be arrived at in a reasonable amount of time... and calculation time is critical, as the parameter space to be explored is usually vast. Therefore a fast computational paradigm is highly advantageous if not absolutely necessary at the present to effectively model neural fields..

When one considers the macroscale, mesoscale, and even microscale...it is interesting to think about computation biologically rather than classically. Biological dynamics are characteristically local. Though evolution of the system is riddled with codes, network graphs, and microprograms, cells are only influenced by a neighborhood of locally surrounding events that dictate the causal response of the particular cell. On one hand, when examining the system as a whole we are faced with astounding complexity...while on the other hand at the cellular level we imagine that the activity of a single cell can be stereotyped into only a relatively small but diverse number of functions. 

A neural network holistically is a complicated distributed system. Information may be processed in parallel, encoded with redundancy and fault tolerance, and manipulated in a vast variety of ways. This distributed network, however, is still locally tied to a neighborhood of events.    The study of such seemingly complicated behavior may possibly be accelerated by reducing it conceptually to an assembly of automata---without risk of reducing the complexity of the resulting behavior.

Ultimately, in bridging the gap between mesoscale and microscale, an understanding of the set of rules, or generative grammar of the single cell must be related to the dynamics of the assembly. Such a system of generative grammar exists in e.g.. the immune system, why not seek a similar characterization of the nervous system? Furthermore, assuming the system is Zipfian, we are guaranteed a minimal grammar, which is a tempting constraint to impose on the problem.

In light of the motivation for such a computational paradigm, a dynamic neural field theory that is implicitly a cellular automation is examined as a first attempt towards unifying micro with macro.

$$ τ·∂_t V(x^μ ,t) = V₀ - V($$x^μ$$ ,t) + θ($$x^μ$$ ,t) + ∫ ∫ dx^ν Γ ($$x^μ$$ ,$$x^ν$$)·f[ V($$x^ν$$ ,t)]

The general form seems to allow an extremely wide class of behavior that has yet to be fully explored. In addition to the dynamics in the literature, this investigation reveals a (new to the author's knowledge) mode that prefers stable, localized oscillations--even when the initial conditions are random or noisy. This mode is a limit-cycle that depends on population size and the decay constant. A localized excitation (high spatial frequency) will spread to a critical radius, at which all solutions tend to the same oscillation frequency. An excitation exceeding the critical radius (low spatial frequency) will completely decay. Random initial conditions therefore naturally settle into both isolated regions of oscillation and regions of inactivity. From here, a slight modification of the automation allows the same oscillators to radiate waves across the manifold, allowing for propagation and interference of signals. Adding a second layer of neurons allows for spiral waves and other complex behavior.  Relating the parameters to physical values remains a difficulty, but a wide variety of relevant phenomena in computational neuroscience may possibly be modeled quickly by such cellular automation.

Evolution From Noise

Stable Oscillator

Dissipative Excitation