This report calls attention to the magnitude and pervasiveness of hysteresis in the coding from length to afferent discharges in crayfish stretch receptor organs (SRO's). The influence of previous lengths on the rate that corresponded to a particular length L was manifest by a substantial excess of that encountered when L was arrived at from a shorter value over that when arrived at from a longer one. Hysteretic loops were present under dynamic conditions when length was modulated quasi-sinusoidally in the length vs. rate Lissajous plots of both the slowly and the fast-adapting organs (SAO, FAO), either not perturbed or perturbed. Loops became narrower with increasing frequency (except for when 1 to 1 locking appeared, Diez Martínez and Segundo, 1983). Hysteretic loops were present under static conditions when length changes were step-like, and fully adapted rates were noted in the SAO and in the perturbed FAO. Earlier reports suggest that hysteresis reflects jointly at least mechanical and electrogenic factors in the "length-to-local dendritic effects" and in the "generator potential to discharge" stages. Several models, either mechanical or mathematical, reveal hysteretic behavior. Detailed analysis has not been performed except for one instance (Chua and Bass, 1972) where, for example, loop-narrowing at higher frequencies occurs only with certain weighting functions whose physiological significance is as yet obscure. Hysteresis may be more widespread than suspected in sensory (and perhaps other) systems: it involves a multi-valuedness that raises the issue of how central mechanisms infer stimulus magnitude retrospectively from the discharge.

No abstract

The correspondence between afferent discharges and sinusoidal length modulations (0.2--10 cps, under 10% of the natural length variations) was studied in isolated fast-adapting stretch receptor organs (FAO) of crayfish, largely using average displays of rate vs. length (or derivatives) along the cycle. Rate modulations were greatest during early cycles and then stabilized, an initial adjustment remindful of mechanical preconditioning. Responses to stimulation in the FAO, as in the slowly-adapting organs (SAO) and possibly other receptors, exhibit the following features, all striking because of their magnitude and ubiquity. i) A zig-zag overall afferent rate vs. stimulus frequency graph with positively and negatively sloped segments. This precludes the straightforward use of Bode plots. ii) Marked non-linearities as an obvious stimulus-response locking in the positively sloped segments, a double-valuedness with one rate while stretching and another while shortening, a lower-limit saturation with the receptor silent for more than half a cycle, and an asymmetric rate sensitivity. iii) Clear-cut discharge leads relative to the stimulus at low frequencies and lags at high ones. The FAO responds worse than the SAO to low frequencies, and better to high ones; it is locked 1-to-1 in a much broader range (e.g., 3--100 vs. 1--3 cps). All features were strongly frequency-dependent. With higher frequencies: i) the number of impulses per cycle fell from several to just one and finally to one every several cycles at higher values; ii) the two values of each length approached one another usually but not always; iii) the silent proportion of the cycle increased; and iv) the rate sensitivity changed. Each feature can arise in principle at any of the transduction stages from length to discharge: the mechanical transduction from length to dendritic deformation, an the encoder one from generator potentials to discharges are particularly likely candidates.

We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of epsilon-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.