FIGURE 8.2-7 The reversed, conditional expectation, Rxy (t) , calculated from a single fiber in a cat auditory nerve. Noise bandwidth was 200 to 2000 Hz; the noise input for 4403 spikes was processed. The peak sinusoidal frequency response for this unit was 1.5 kHz. (From DeBoer, E. and Kuyper, P., IEEE Trans. Bio. Med. Eng., 15(3): 169, 1968. © 1968 IEEE. With permission from IEEE.)
Taking the inverse LaPlace transform of H(s) yields its weighting function or impulse response, h(t):
h(t) = ——--e-at sin(bt +9), where 9= tan-1(b/-a) 8.2-44
In terms of the more familiar system parameters, Q and ran, h(t) can be written:
h(t) ^ exp[((ön/2Q)t]sin[ffl J(l-l/4Q2) t + 9] 8.2-45
where: a = ran/2Q, b = ran (1-1/4Q2), ran is the undamped natural frequency of the filter, Z is the damping factor of the quadratic pole pair, and Q = 1/2Z. Note that as the Q increases, the damping envelope on the sinusoid is sustained. That is, there are more cycles visible in h(t), and it has a longer settling time.
Wu (1970) applied TC to both electronic models and the auditory system of the grasshopper, Romalea microptera. Romalea has paired, external, tympanal organs on the sides of the first abdominal segment. The tympanic membrane is oval, measuring about 3.6 mm long by 2 mm wide. A tympanic ganglion with 60 to 70 sensillae is attached to the inner surface of the tympanic membrane. The auditory nerves run from the tympanic ganglia to the third thoracic ganglion. Wu obtained single-unit auditory responses with microelectrodes inserted into a tympanic ganglion. Both steady-state sinusoidal acoustic excitation was used, as well as broadband Gaussian noise. Wu found that typical units responded to the sinusoidal stimuli with relatively broad bandpass characteristics. A typical unit had its peak (0 dB) response at 3 kHz. From the width of its frequency response curve, this unit had a Q = 1.15, or an equivalent quadratic damping factor of Z = 0.43. These numbers suggest that the equivalent h(t) would have one major peak followed by one small undershoot. Wu found that under broadband acoustic noise stimulation, for every unit studied, no significant x+ (t) of any shape emerged from the computational noise. Using the same recording apparatus and calculation techniques, and an analog electronic model of the bandpass filter, Wu did find a well-defined x+(t) of the expected shape for the low-Q condition, however.
One explanation for the lack of results was that the auditory sensillae did not fire in a 1:1 manner with the applied sinusoidal frequency. This behavior suggested that coherent, mechanical stimulation by a number of cycles at a given frequency were required for a sensillum to fire. Under broadband noise stimulation, the tym-panal membrane may have jumped randomly from one mechanical resonance mode to another, never providing enough sustained cycles of mechanical stimulation of the sensory neuron to preserve coherent causality.
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