TY - JOUR
T1 - Closed-loop stabilization of the Jamming Avoidance Response reveals its locally unstable and globally nonlinear dynamics
JF - The Journal of Experimental Biology
JO - J. Exp. Biol.
SP - 4272
LP - 4284
DO - 10.1242/jeb.088922
VL - 216
IS - 22
AU - Madhav, Manu S.
AU - Stamper, Sarah A.
AU - Fortune, Eric S.
AU - Cowan, Noah J.
Y1 - 2013/11/15
UR - http://jeb.biologists.org/content/216/22/4272.abstract
N2 - The Jamming Avoidance Response, or JAR, in the weakly electric fish has been analyzed at all levels of organization, from whole-organism behavior down to specific ion channels. Nevertheless, a parsimonious description of the JAR behavior in terms of a dynamical system model has not been achieved at least in part due to the fact that ‘avoidance’ behaviors are both intrinsically unstable and nonlinear. We overcame the instability of the JAR in Eigenmannia virescens by closing a feedback loop around the behavioral response of the animal. Specifically, the instantaneous frequency of a jamming stimulus was tied to the fish's own electrogenic frequency by a feedback law. Without feedback, the fish's own frequency diverges from the stimulus frequency, but appropriate feedback stabilizes the behavior. After stabilizing the system, we measured the responses in the fish's instantaneous frequency to various stimuli. A delayed first-order linear system model fitted the behavior near the equilibrium. Coherence to white noise stimuli together with quantitative agreement across stimulus types supported this local linear model. Next, we examined the intrinsic nonlinearity of the behavior using clamped frequency difference experiments to extend the model beyond the neighborhood of the equilibrium. The resulting nonlinear model is composed of competing motor return and sensory escape terms. The model reproduces responses to step and ramp changes in the difference frequency (df) and predicts a ‘snap-through’ bifurcation as a function of dF that we confirmed experimentally. AICAkaike information criterionAMamplitude modulationBICBayesian information criterionD(s)Laplace transform of difference signald(t)difference signal (Hz)dfdifference frequency (Hz)e(d)escape functionEODelectric organ dischargef1EOD frequency (Hz)f2stimulus frequency (Hz)FRFfrequency response functionG(s)behavior transfer functionH(s)closed-loop transfer functionJ(s)open-loop transfer functionJARJamming Avoidance ResponseKnumber of parameters in candidate modelktransfer function gain (Hz)M(s)model transfer functionNnumber of fishnnumber of trialsndexistence of delay (0/1) in candidate modelnpnumber of poles in candidate modelnznumber of zeros in candidate modelppole of transfer function (Hz)R(s)Laplace transform of reference signalr(t)reference signal (Hz)RRresponse–response coherencesLaplace complex frequency variableSRstimulus–response coherenceTdelay of transfer function (s)ttime (s)U(s)Laplace transform of input signalu(t)input signal (Hz)Y(s)Laplace transform of output signaly(t)output signal (Hz)αfeedback gainρ(y)return functionτtime constant (s)ωfrequency (rad s−1)
ER -