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Synapses
Coherent brain waves are generated by clusters of neural circuits.
Individual neurons in neural circuits synapse in ways which determine
how groups of circuits synchronize with each other. In developing
a basis for neural computation, I wanted to use a formal mathematical
method to describe how the synapses of individual neurons could
generated the synchronous signals in neural circuits. The formal
method in modern signal processing of using the Gabor wave packet
transforms is a general method for describing neural signals.
The measurement of a single spike is subject to the uncertainty
principle. The temporal width of the spike is T, and its
sharpness is dependent on the neuron's energy output. The
larger the energy of the spike the narrower the shape of the
spike.
A neuron usually communicates with other neurons by oscillating
repeatingly, synapsing rapidly many times within a relatively
small time window T. The wave packet is created when a group of spikes
all near the same frequency and phase clump together. The wave packet
represents a region where a localized concentration of energy occurs.
The basis for this occurrence can be understood by an approximation
called the linear superposition principle of waves. In theory, no waves
in the real world are perfectly linear due to the uncertainty principle.
Furthermore, it is widely believed that waves only occur in a
"many-body" media or ensembles in space-time. The ellisodial
waves on the ocean, the longitudinal sound waves from the birds, and
the transverse electromagnetic waves in the light from the sun and stars
all need a medium to travel through. This medium is dynamic with
energy, not an empty abstract void.
When waves overlay in frequency and phase, they are said to be
synchronized. This synchonization is seen in the electroencephalographic
(EEG) recordings taken off the scalp of the brain. Synchronized,
coherent neural circuit paths emerge when the synapses fire uniformly
as cascaded spike-trains. It's intuitively likely that neural circuits,
like electronic circuits can be tuned to resonate to distinct
wave pulses. The simple parallel RC-L circuit can be tuned by adjusting
the capacitance C [1]. The resonance frequency in this simple circuit is
a function of maximum circuit impedance which is roughly inversely
proportional to the capacitance C. So a rising capacitance in a low
resistive, energy efficient, neural circuit will tend to keep the circuit
response frequency in a narrow band and pull the median circuit frequency
down.
References
1. The following is an excerpt from:
Resonance, Oscillation and the Intrinsic Frequency Preferences of
Neurons
Bruce Hutcheon and Yosef Yarom
Trends in Neuroscience (2000). 23, 216
"There is no difficulty in locating which properties of neurons
result in low-pass filtering characteristics. The mechanism is well
known and ubiquitous. It is a fundamental property of all cells
that the parallel leak conductance and capacitance of the outer
membrane forms the equivalent of a filter that attenuates responses to
inputs at high frequencies."
"There are two elementary rules for deciding which voltage-gated currents
will act as high-pass filters and will therefore be capable of combining
with the passive properties of neurons to produce resonance.
(1) Currents that actively oppose changes in membrane voltage can
produce resonance.
(2) To produce resonance, currents that meet the criterion above must,
in addition, activate slowly relative to the membrane time constant."
PDF file - Hutcheon-Yarom
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Coherence
