Home-> Computation->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.


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

next: Coherence