Unidirectional Synchronization of Hodgkin-Huxley Neurons With Prescribed Performance Under Transcranial Magneto-Acoustical Simulation

Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a "physiological state diagram," with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.

The development of orderly connections in the mammalian visual system depends on action potentials in the optic nerve fibers, even before the retina receives visual input. In particular, it has been suggested that correlated firing of retinal ganglion cells in the same eye directs the segregation of their synaptic terminals into eye-specific layers within the lateral geniculate nucleus. Such correlations in electrical activity were found by simultaneous recording of the extracellular action potentials of up to 100 ganglion cells in the isolated retina of the newborn ferret and the fetal cat. These neurons fired spikes in nearly synchronous bursts lasting a few seconds and separated by 1 to 2 minutes of silence. Individual bursts consisted of a wave of excitation, several hundred micrometers wide, sweeping across the retina at about 100 micrometers per second. These concerted firing patterns have the appropriate spatial and temporal properties to guide the refinement of connections between the retina and the lateral geniculate nucleus.