Localized electrical nerve blocking was investigated in computer simulation and in vivo trials for sinusoidal frequencies between 5 and 20 kHz. Computer simulations indicated that a localized transmission block of the axons could occur in each of the axon models. An approximation of nerve stimulation was derived from individual axon simulations conducted over axon diameters of 5-15 microm and electrode to axon distances of 0.25 to 2.0 mm. Examination of the membrane voltage and ionic gate potentials indicated that the block could be attributed to an elevated membrane voltage. The elevated membrane voltage could prevent conduction of action potentials through the region of the sinusoidal currents. At lower amplitudes, the sinusoidal current could stimulate the axon and generate a continuous series of action potentials. In vivo trials demonstrated that the sinusoidal frequencies of greater than 10 kHz would cause a localized block in rats. Sinusoidal frequencies below 5 kHz would lead to a reduction in muscle force that appeared to be caused by depletion of transmitter at the neuromuscular junction. As indicated by the computer models of rat nerves, the endplate depletion block occurred at a lower frequency (below 5 kHz) than the block (above 10 kHz). A partial block of the axon was demonstrated, suggesting that sinusoidal currents could be used to provide selective stimulation if they are combined with distal electrical stimulation.
The case m -2k appears to be easier to handle, since the obstruction is the intersection form, which is just the ®-product of the form on M and the form on N, and is homologically defined. The self intersection form does introduce a complication, at least if k is odd. In any case we concentrate here on m = 2k + l.
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