The fast transient potassium or A current (IA) plays an important role in determining the activity of central pattern generator neurons. We have previously shown that the shal K+ channel gene encodes IA in neurons of the pyloric network in the spiny lobster. To further study how IA shapes pyloric neuron and network activity, we microinjected RNA for a shal-GFP fusion protein into four identified pyloric neuron types. Neurons expressing shal-GFP had a constant increase in IA amplitude, regardless of cell type. This increase in IA was paralleled by a concomitant increase in the hyperpolarization-activated cation current Ih in all pyloric neurons. Despite significant increases in these currents, only modest changes in cell firing properties were observed. We used models to test two hypotheses to explain this failure to change firing properties. First, this may reflect the mislocalization of the expressed shal protein solely to the somata and initial neurites of injected neurons, rendering it electrically remote from the integrating region in the neuropil. To test this hypothesis, we generated a multicompartment model where increases in IA could be localized to the soma, initial neurite, or neuropil/axon compartments. Although spike activity was somewhat more sensitive to increases in neuropil/axon versus somatic/primary neurite IA, increases in IA limited to the soma and primary neurite still evoked much more dramatic changes than were seen in the shal-GFP-injected neurons. Second, the effect of the increased IA could be compensated by the endogenous increase in Ih. To test this, we modeled the compensatory increases of IA and Ih with a cycling two-cell model. We found that the increase in Ih was sufficient to compensate the effects of increased IA, provided that they increase in a constant ratio, as we observed experimentally in both shal-injected and noninjected neurons. Thus an activity-independent homeostatic mechanism maintains constant neuronal activity in the face of dramatic increases in IA.

Subjective tinnitus is a phantom sound sensation that does not result from acoustic stimulation and is audible to the affected subject only. Tinnitus-like sensations in animals can be evoked by procedures that also cause tinnitus in humans. In gerbils, we investigated brain activation after systemic application of sodium salicylate or exposure to loud noise, both known to be reliable tinnitus-inductors. Brains were screened for neurons containing the c-fos protein. After salicylate injections, auditory cortex was the only auditory area with consistently increased numbers of immunoreactive neurons compared to controls. Exposure to impulse noise led to prolonged c-fos expression in auditory cortex and dorsal cochlear nucleus. After both manipulations c-fos expression was increased in the amygdala, in thalamic midline, and intralaminar areas, in frontal cortex, as well as in hypothalamic and brainstem regions involved in behavioral and physiological defensive reactions. Activation of these non-auditory areas was attributed to acute stress, to aversive-affective components and autonomous reactions associated with the treatments and a resulting tinnitus. The present findings are in accordance with former results that provided evidence for suppressed activation in auditory midbrain but enhanced activation of the auditory cortex after injecting high doses of salicylate. In addition, our present results provide evidence that acute stress coinciding with a disruption of hearing may evoke activation of the auditory cortex. We interpret these results in favor of our model of central tinnitus generation.

Abstract. The Hodgkin-Huxley model was developed to characterize the action potential of a squid axon. It has served as an archetype for compartmental models of the electrophysiology of biological membranes. Thus the dynamics of the Hodgkin-Huxley model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods that can be applied to more complex models. This note demonstrates previously unobserved dynamics in the Hodgkin-Huxley model, namely, the existence of chaotic solutions in the model with its original parameters. The solutions are found by displaying rectangles in a cross-section whose images under the return map produce a Smale horseshoe. The chaotic solutions are highly unstable, but they are significant as they lie in the basin boundary that establishes the threshold of the system.

Abstract. This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1-35].The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1-35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system.

The hyperpolarization-activated cation current (Ih) is widely distributed in excitable cells. Ih plays important roles in regulation of cellular excitability, rhythmic activity, and synaptic function. We previously showed that, in pyloric dilator (PD) neurons of the stomatogastric ganglion (STG) of spiny lobsters, Ih can be endogenously upregulated to compensate for artificial overexpression of the Shal transient potassium channel; this maintains normal firing properties of the neuron despite large increases in potassium current. To further explore the function of Ih in the pyloric network, we injected cRNA of PAIH, a lobster gene that encodes Ih, into rhythmically active PD neurons. Overexpression of PAIH produced a fourfold increase in Ih, although with somewhat different biophysical properties than the endogenous current. Compared with the endogenous Ih, the voltage for half-maximal activation of the PAIH-evoked current was depolarized by 10 mV, and its activation kinetics were significantly faster. This increase in Ih did not affect the expression of IA or other outward currents. Instead, it significantly altered the firing properties of the PD neurons. Increased Ih depolarized the minimum membrane potential of the cell, reduced the oscillation amplitude, decreased the time to the first spike, and increased the duty cycle and number of action potentials per burst. We used both dynamic-clamp experiments, injecting the modeled PAIH currents into PD cells in a functioning STG, and a theoretical model of a two-cell network to demonstrate that the increased Ih was sufficient to cause the observed changes in the PD activity.