All sensory receptor cells are transducers; they respond to the physical quantity under measurement (QUM) by a change in their transmembrane potential, Vm. In most cases, Vm goes positive (depolarization) as a nonlinear, monotonic function of the QUM. Depolarization of Vm generally leads to the production of nerve spikes at the spike generator locus (SGL) of the cell. However, some receptor cells do not produce spikes; their depolarization is coupled by either electrical or chemical synapses to a sensory interneuron that does spike. Certain receptors, such as vertebrate rods and cones, hyperpolarize with increasing stimulus intensity (absorbed light power) and affect a complex of signal processing cells in the retina, the underlying neuropile in the vertebrate eye. The outputs of the retina are spikes on the axons of the ganglion cells that form the optic nerve, which carries spike signals to the CNS.

Still other receptors, such as the photoreceptors in the eye of the plecypod mollusk, Mytilus edulis, respond to light only at OFF. That is, they fire a burst when the illumination dims or goes off, and do not fire for ON or brightening at all (see Section 2.9).

Most receptors that respond positively to increasing QUM do so in a manner that suggests that the QUM is acted on by physical processes in which the receptor's spike frequency is driven by an approximate proportional plus derivative operation on the QUM as a function of time. That is, at QUM ON, the instantaneous spike frequency of the receptor jumps to a peak value, then slowly declines to a lower steady-state value, or even to zero. This proportional plus derivative-like response is called adaptation of response by neurophysiologists. Often, the spike instantaneous frequency (IF) of certain adapting receptors in response to a step ON of the QUM is like that of a linear system where r(t) = Ae-at + B. There may be more than one exponential term, so, for example, r(t) = Ae-at + Be-||t + C. In this latter case, the peak IF is (A + B + C), and the steady-state IF is C pps.

The IF at ON for other receptors may be fit better by a plot of linear frequency vs. log(time) coordinates. That is, the step response IF is fit by a mathematical model of the form: r(t) = Kt-P. After Laplace transformation r(t) and multiplication of this transform by s, the impulse response of the receptor is H(s) = K T(1 - |)sp, where r(x) is the gamma function. In one example in Milsum (1966), a cockroach mech-anoreceptor was described with K = 23 and | = 0.76.

Very often the response of a receptor exhibits unidirectional rate sensitivity. That is, the strong derivative component in the step (ON) response is lacking in the OFF response. This may be true even if the receptor has a steady-state firing rate, ro, for a zero-QUM. The firing rate drops quickly to ro with no undershoot to zero. That ON and OFF response dynamics differ is probably due to the fact that the physical/chemical processes mediating the ON response are different from those involved with the OFF dynamics.

A very rapidly adapting receptor (e.g., the pacinian corpuscle) fires a short burst of spikes at ON of the QUM (pressure), and another short burst at OFF. In other words, the changes in pressure are important to these sensors. A conceptual model for this sensor is shown in Figure 2.1-1. Note that the absolute value of the derivative term is used to describe the ON and OFF response of this receptor.

2.1.2 Receptor Nonlinearity

Many receptors can operate at the theoretical limit of signal detection, and have an enormous dynamic range. In an engineering context, a large dynamic range can be obtained by using a nonlinear, gain compression-type nonlinearity, such as using the logarithm of the QUM at the front end of the receptor. For certain neuro-sensory systems, it is speculated that the CNS sends an efferent signal that adjusts the SGL firing threshold of the receptor so that, in the absence of the QUM, the threshold is made very low, such that the receptor is maximally sensitive, occasionally firing on membrane noise. In the presence of a large QUM, the sensor spike output causes the CNS to send an efferent signal to the receptor to raise its threshold so that the output spike rate does not saturate. (Few receptors can fire over 500 pps because of basic nerve membrane dynamics.)

Many receptors exhibit a log/linear transfer characteristic over a sizable potion of their dynamic range, both for their initial firing rate at ON, and for the rate sampled later after adaptation has occurred. That is:

where r is the IF of the receptor, Kr and ro are positive constants, and Io is the intensity threshold. This relation is basically the mathematical result of the Weber-Fechner law for perception which says that the just noticeable difference in a QUM is proportional to (I - Io) (Milsum, 1966). Another mathematical model for the perceived output of receptors comes from Stevens (1964). Stevens noted that the perceived intensity of a QUM, Y, can be modeled by

The exponent, v, ranges between 0.33 and 3.5, the larger values generally associated with noxious QUMs such as electroshock, heat, or pain. Of course, r and Y are nonnegative quantities.

If the full dynamic range of a typical receptor, e.g., a touch receptor, is examined, the output frequency actually has four ranges with respect to the intensity of the QUM. These are shown in Figure 2.1-2. At QUM intensities ranging from zero to I9, r = 0. This is in effect a dead zone. From I9 to Io, r increases with a slope less than K log(I - Io), and in the range Io 3 I 3 Iu, r is approximated by K log(I - Io). For I > Iu, r flattens out and saturates at rmax. The dead zone and the low slope range (0 < I < Io) give the receptor a certain robustness against noise. As mentioned above, certain receptors fire slowly and randomly for I = 0. That is, a certain noisiness in their spike output is tolerated to obtain enhanced sensitivity to very low I > 0.

As is the case with all physical measurements, the ultimate limit to detectability of the QUM is noise arising in the receptor as well as noise accompanying the QUM

Receptor output frequency

Receptor output frequency

QUM intensity

FIGURE 2.1-2 Graph showing the instantaneous output frequency of a typical receptor's vs. stimulus intensity. (Note that some receptors do not generate output spikes, but rather a generator potential.) If this curve represents the instantaneous frequency (IF) at ON of a step stimulus, the steady-state IF vs. stimulus intensity will have a similar shape, but a much lower slope. See text for discussion.

FIGURE 2.1-2 Graph showing the instantaneous output frequency of a typical receptor's vs. stimulus intensity. (Note that some receptors do not generate output spikes, but rather a generator potential.) If this curve represents the instantaneous frequency (IF) at ON of a step stimulus, the steady-state IF vs. stimulus intensity will have a similar shape, but a much lower slope. See text for discussion.

(environmental noise). Assume for now that the environmental noise is zero, that is, the signal-to-noise ratio (SNR) at the input is infinite. Now the smallest detectable signal depends on the inherent noisiness of Vm with QUM = 0. Noise in Vm can come from several sources: (1) The random leakage of ions (e.g., Na+ inward, K+ outward, etc.) through the cell membrane specific ion channel proteins. (2) Thermal (Johnson) noise current arising from the bulk, resting, membrane conductance. This is given by: :inm = 4kTGm B mean squared amp/cm2. B is the hertz bandwidth over which the noise is viewed, and Gm is the bulk membrane conductivity in S/cm2. If the thermal noise plus leakage noise [vn (t)] plus the resting potential (Vmr) exceeds the sensory neuron SGL firing threshold (V9), it will fire, giving a false-positive output. A false negative can occur as well when QUM > 0 so that Vm would normally exceed the firing threshold, but [Vmr + vn (t)] < Vr Now when vn (t) goes positive, the receptor will fire but at a higher rate than if vn = 0.

One of the challenges in sensory neurophysiology is to try to understand how an animal detects the change in the random firing pattern of a sensory neuron axon from the zero-QUM condition to the firing statistics present when there is a threshold level of depolarization caused by a nonzero, threshold QUM intensity. (The probabilistic approach to threshold sensory perception is treated rigorously by Reike et al. (1997) in their book on the neuro-sensory code, Spikes.)

The use of efferent feedback to optimize or maximize detection probability while minimizing false-positive spikes was mentioned above. It is worth noting that many neuro-sensory systems capable of great sensitivity are known to have efferent fibers in their nerves. These include both vertebrate "camera" eyes and arthropod compound eyes, the vertebrate cochlea, the statocysts of octopus, etc.

This section, using basic probability theory, derives conditions for an optimum SGL firing threshold that will minimize the operating "cost" of a sensory neuron, defined below. The firing threshold is a fixed, positive voltage defined by

Assume that a bandwidth-limited Gaussian noise, vn, is added to Vmr. vn is defined to have zero mean, E[vn] = 0, and a variance, oI = e{v;;} . The derivative of vn also has zero mean and variance, od = E-jv;;} . Another required property of the membrane voltage noise autocorrelation function, Rm(x) is dRnn (t) = 0, for t = 0 2.1-4

Assume that the QUM occurs randomly as short, infrequent pulses of amplitude A. For an input pulse to cause the sensory neuron to fire, d vn(t)/dt > 0, and [vn(t) + AP(t)] S vo. P(t) is defined as a pulse of short width, e s, and peak height 1. When the neuron fires in response to an input pulse, the event is called a true positive; its firing rate due to AP(t) is RTP. From probability theory, RTP = X Pr(yo 3 (vn + A) < (vo + A)}. X is the mean rate of occurrence of input pulses. Because it is assumed that vn is described by a Gaussian probability density function, it is easy to show that

where erf(z) is the error function, defined as erf (z) = (2A/n) exp(-t2 )dt 2.1-6

Note that erf(0) = 0, and erf(x) = 1. Two types of error can occur for this type of threshold event detector. A false positive, where an output spike occurs because vn has crossed the firing threshold with positive slope, giving a rate of false-positive (RFP) outputs. The second type of error is a false negative, where the instantaneous sum (vn + A) < vo, or vn > yo. The RFP is given by (Papoulis, 1965):

RFP is the mean rate vn crosses with positive slope. The rate of false negatives (RFN) is given by

Again, by integrating probability densities,

RFN = (V2)|l + erf[(o - A))onV2)]} + (X/2){1-erf[(¥o)(o^V2)]} 2.1-9

Now an operating cost is defined for the receptor that will be minimized:

C = (1/rc)(o Jon) exp[-¥ ^/(2o 2)] + X + Xerf[( o - A))(o)]

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