Control of time-dependent biological processes by temporally patterned input

Abstract

Temporal patterning of biological variables, in the form of oscillations and rhythms on many time scales, is ubiquitous. Altering the temporal pattern of an input variable greatly affects the output of many biological processes. We develop here a conceptual framework for a quantitative understanding of such pattern dependence, focusing particularly on nonlinear, saturable, time-dependent processes that abound in biophysics, biochemistry, and physiology. We show theoretically that pattern dependence is governed by the nonlinearity of the input-output transformation as well as its time constant. As a result, only patterns on certain time scales permit the expression of pattern dependence, and processes with different time constants can respond preferentially to different patterns. This has implications for temporal coding and decoding, and allows differential control of processes through pattern. We show how pattern dependence can be quantitatively predicted using only information from steady, unpatterned input. To apply our ideas, we analyze, in an experimental example, how muscle contraction depends on the pattern of motorneuron firing.

Figure 1

Pattern and pattern dependence: definitions and examples. (A) Specification of the input waveforms used throughout this work. t, time; i, input amplitude; 〈i〉, mean i; P, period; F, duty cycle; ℐ₀, unpatterned waveform of steady input at i = 〈i〉; ℐ_x, a patterned waveform of three times higher input for one-third of the time. (B) Dependence of contraction of the ARC muscle of Aplysia on (C) the pattern of motorneuron firing. Experiments were done as in (28, 29). Motorneuron B15 [(26); B15 was used in all experiments presented in this paper; results with B16 were similar] was intracellularly stimulated to fire spikes [individual spikes were driven by separate brief current injections (28), not shown] in the desired pattern, always of the form in A. The firing frequency was taken as the input variable i and contraction amplitude as the output variable o (A13). Contractions were isotonic and unloaded; length was monitored with an isotonic transducer. ō, peak contraction; 〈o〉, mean contraction. With steady, unpatterned firing (waveform ℐ₀), the contraction reaches the steady state ō(ℐ₀) = 〈o〉(ℐ₀) (Left); with patterned firing (waveforms ℐ_x), it reaches different values ō(ℐ_x) and 〈o〉(ℐ_x), as indicated for the rightmost pattern. By Eq. 2, normalizing the values for ℐ_x by that for ℐ₀ establishes the scale of absolute pattern dependence Φ, shown on the right of B.

Figure 2

Theoretical properties of pattern dependence, with transformation f given by schema 3. (A–C) For three representative cases, Φ_ō (Left) and Φ_〈o〉 (Right) are plotted as functions of the pattern (F, P), using Eqs. 6 and 7 in A9. Coordinates as well as locations of the line P ≈ τ (A8) and surface Φ = 1 are indicated in the upper middle diagram. Φ = 1 is represented in medium gray, Φ > 1 lighter, and Φ < 1 darker. In all cases α,β = 1, p and q are as indicated, and 〈i〉 = 0.01 (A) or 0.25 (B and C). These values of 〈i〉 were chosen to give o_∞(〈i〉) ≈ 0.01 and τ ≈ 1 in all three cases. (D) Linear plot of the circled region in C. (E) The steady-state f: o_∞ − i relations for the three cases in A–C (A10). For P ≫ τ, Φ_ō and Φ_〈o〉 may be computed from such relations as follows. As F3 shows, ō(ℐ_x) = o_∞(i_x) and 〈o〉(ℐ_x) = Fo_∞(i_x), and of course ō(ℐ₀) = 〈o〉(ℐ₀) = o_∞(〈i〉). Then, from Eq. 2, Φ_ō = o_∞(i_x)/o_∞(〈i〉) and Φ_〈o〉 = Fo_∞(i_x)/o_∞(〈i〉 = Fφ_ō). Setting Φ_ō = 1 and Φ_〈o〉 = 1 yields equations for the two lines shown. For any particular 〈i〉 and F [here illustrated for 〈i〉 on curve B and F = 0.5, so that i_x = 2〈i〉], defining the points 1–4 as shown and writing o_∞,1 for the o_∞ value at point 1, etc., we see that Φ_ō = o_∞,2/o_∞,1 = o_∞,2/o_∞,4 and φ_〈o〉 = Fo_∞,2/o_∞,1 = o_∞,2/o_∞,3. (Φ is much smaller here than in A–C because 〈i〉 is much larger.) Φ can thus be computed even from a purely empirical o_∞−i relation. In this theoretical case, of course, we know the precise equivalent general expressions for Φ (A10). (F) Time courses of o at the three locations indicated in A—i.e., the solutions (A9) of schema 3 (Eq. 5) with α, β, p, q = 1 (since q = 1, these are also the time courses of a), for the input waveforms ℐ₀ and ℐ_x with 〈i〉 = 0.01, F = 1/3, i_x = 3〈i〉 (Bottom), when P ≪ τ (F1), P ≈ τ (F2), and P ≫ τ (F3).

Figure 3

Experimental analysis of pattern dependence in the Aplysia ARC-muscle system. (A and B) Contraction kinetics and o_∞−i relations (from the experiment in B and two others) obtained with steady, unpatterned motorneuron firing (schematically indicated under B). To summarize the data for the purposes of computation, we fitted the kinetics, at each i, with a delay d(i), then a single-exponential rise with time constant τ(i), to the steady-state amplitude o_∞(i) (thin, smooth curves in B). Though the actual kinetics are clearly more complex, this form appeared to provide the most simple yet reasonably adequate empirical description of the data. (C) Φ_ō and Φ_〈o〉 predicted from the values obtained from A and B (mesh; computed using the same strategy as in A9) and experimentally measured (scatter points; steady-state measurements as in Fig. 1 B and C). The measured values are from 11 preparations, which gave surfaces of Φ of similar shape but different absolute amplitude. The values from each preparation were therefore scaled so as to normalize the reference value Φ(F = 0.5, P = 10 s) to the mean from all preparations (Φ_ō, 27 ± 19 SD, range 13–75; Φ_〈o〉, 14 ± 9 SD, range 6–37). The mean 〈i〉 was 5.7 Hz (±1.2 SD, range 4–8); the same value was used in the theoretical computations. The mean deviation of the experimental points from the predicted surface is 3.4 for Φ_ō, 1.0 for Φ_〈o〉 (n = 132, values for F = 1 excluded). (D) Cooling the preparation (here from 20.3 to 14.9°C; unpatterned firing at 4 Hz) greatly increases contraction amplitude. (E) Concomitantly, cooling greatly reduces Φ. Means ± SEM from four preparations (not normalized) first at 20–21.1°C, then 14.9–15°C. 〈i〉 = 4–5.5 Hz; F = 0.25. [The experiments in Figs. 1 B and C and 3 A–C were done at the “warm” temperatures. All temperatures used are in the physiological range for Aplysia (40).]

Figure 4

Different processes can be tuned to respond to different patterns of the same input variable. This is, essentially, a section through the φ_〈o〉 surface in Fig. 2C (thus p = 1, q = 3, 〈i〉 = 0.25) at F = −3 for three processes with time constants two orders of magnitude apart: process 1, α = 400, β = 1,000; process 2, α = 4, β = 10; process 3, α = 0.04, β = 0.1. “Windows” of pattern dependence such as these are observed experimentally (11, 19, 20).