The effects of temperature on the stability of a neuronal oscillator

Abstract

The crab Cancer borealis undergoes large daily fluctuations in environmental temperature (8-24°C) and must maintain appropriate neural function in the face of this perturbation. In the pyloric circuit of the crab stomatogastric ganglion, we pharmacologically isolated the pacemaker kernel (the AB and PD neurons) and characterized its behavior in response to temperature ramps from 7°C to 31°C. For moderate temperatures, the pacemaker displayed a frequency-temperature curve statistically indistinguishable from that of the intact circuit, and like the intact circuit maintained a constant duty cycle. At high temperatures (above 23°C), a variety of different behaviors were seen: in some preparations the pacemaker increased in frequency, in some it slowed, and in many preparations the pacemaker stopped oscillating ("crashed"). Furthermore, these crashes seemed to fall into two qualitatively different classes. Additionally, the animal-to-animal variability in frequency increased at high temperatures. We used a series of Morris-Lecar mathematical models to gain insight into these phenomena. The biophysical components of the final model have temperature sensitivities similar to those found in nature, and can crash via two qualitatively different mechanisms that resemble those observed experimentally. The crash type is determined by the precise parameters of the model at the reference temperature, 11°C, which could explain why some preparations seem to crash in one way and some in another. Furthermore, even models with very similar behavior at the reference temperature diverge greatly at high temperatures, resembling the experimental observations.

Figure 1. The pyloric network, intact and with pacemaker isolated.

(A) At top, a schematic of the intact pyloric network. Dots represent inhibitory chemical synapses, resistor symbols indicates electrical synapses. Dot color represents the transmitter used by particular synapses. ACh = acetylcholine, Glu = glutamate. At bottom, intracellular recording of PD and simultaneous extracellular recordings from three nerves: pdn, gpn, and pyn, which reflect activity in PD, LP, and PY, respectively. Traces recorded at 11°C. (B) At top, a schematic of the pyloric network in presence of 10⁻⁵ M PTX, which blocks glutamatergic synapses in C. borealis. In this condition, the major synaptic input to the pacemaker from other pyloric neurons has been blocked. At bottom, the same preparation as in panel A, but after application of 10⁻⁵ M PTX. Deprived of pacemaker input, LP and PY fire tonically, but the pacemaker continues to oscillate.

Figure 2. Effect of temperature on frequency and duty cycle of the isolated pyloric pacemaker.

(A) Membrane potential of the PD neuron in one animal, at several temperatures. (B) Similar to A, but in a different individual. (C) PD burst frequency versus temperature for n = 14 individuals (gray), and averaged across individuals (black). Gray lines that end in a dot represent animals that crashed above that temperature. Error bars represent SD. (D) The coefficient of variation of the burst frequency (across individuals) at each temperature, and the variance of the log-transformed frequency (S² _log, see Methods). Brackets denote a Levene s test to compare the log-transformed variance between two temperatures: n.s. = not significant, ** = p<0.01. (E) Pyloric frequency versus temperature, for the isolated pacemaker (black, subset of the data shown in panel C), and the intact network (gray, n = 15, as previously reported in [14], [15]). Data for 31°C is not shown because many preparations crashed or cycled erratically at this temperature. Error bars represent SD. (F) Duty cycle versus temperature, again in the isolated pacemaker (black, n = 12) and the whole network (gray, n = 15, as previously reported in [14], [15]). Again, data for 31°C is not shown. Error bars represent SD.

Figure 3. Examples of isolated pacemaker crashes at high temperatures.

(A) Simultaneous intracellular recording from two PD neurons, and an extracellular recording from the pdn, as temperature is dropped from 27 to 23°C. In the top trace, spikes were cut off at −40 mV. (B, C) Two examples of similarly behaving oscillations: as temperature was increased, from around 18 to 30°C, amplitude dropped and frequency continued to increase, eventually terminating in small-amplitude oscillations. (D) PD recording at a steady 35°C shows multiple switching between oscillation and flat-line voltage seemingly through fold limit cycle bifurcations.

Figure 4. Difference in model channel temperature dependencies produces bifurcation at high temperatures.

Waveforms and corresponding phase plots are plotted as examples for two different Q₁₀ relationships: (A) and (B) , . Phase plots have the gating variable n on the y-axis and the voltage on the x-axis. Thin green line is the V nullcline (the line where ) and the thin blue line is the n nullcline (where ). All red lines correspond to the duty cycle threshold line, chosen as the inward half activation voltage (−50 mV). Thickest black line is the limit cycle. Black dot is a stable fixed point. Lower panel plots capture simultaneous frequency, amplitude, and duty cycle plotted from 0 to 35°C with reference temperature of 11°C. Each point is calculated from the steady state solution of the model equations.

Figure 5. Generalized parameter scaling maps.

Values from the reference model are plotted without scaling at  = 1 and  = 1. From the given reference model, along the x-axis, all conductances () are log-scaled together; on the y-axis, the gating variable () is log-scaled independently. Each point on the color plot corresponds to the measurements from the steady state model ran at their respective scaling factors. The maps of model outputs plot frequency (A), amplitude (B), and duty cycle (C). The dark blue region represents parameters where no oscillations exist. The diagonal (unity) line corresponds to a slice through parameter space where , as in Figure 4A; the white line with a slope of 2 corresponds to the parameter space from Figure 4B.

Figure 6. Variation in Q₁₀ ratios achieves high temperature variability in model output.

Temperature dependencies in the model are as follows:  =  = 1.5,  = 1.6,  = 3. Each grid point corresponds to a model output at steady state. Temperature is plotted from 0 to 45°C, is varied from 0.04 to 0.09 µS. The maps of model outputs plot frequency (A), amplitude (B), and duty cycle (C). The amplitude map points to domains where the transition to instability happens through a supercritical Hopf or a fold limit cycle bifurcation. The mark indicates where these two bifurcations coalesce. The two white lines (1, 2) are chosen as two representative curves with qualitatively different behaviors. Line 1 is at  = 0.07 µS and line 2 is at  = 0.051 µS. The waveform, phase plots and scaling behavior (frequency, amplitude, duty cycle) are plotted for each line to show specific examples. For the phase plots, the thin green line is the V nullcline (where ) and the thin blue line is the n nullcline (where ). Red line is the duty cycle threshold line chosen as the inward half activation voltage (−50 mV). Black line is the limit cycle.

Figure 7. Restricting low temperature frequency output constraints high temperature variability.

Three models were generated with the same reference parameters, but different values. Temperature dependence fixed at:  =  = 1.5,  = 1.6,  = 3. Frequency, amplitude and duty cycle maps were generated (as in Figure 5) for the three parameter points, varying temperature and inward conductance: (A)  = 0.1 µS, (B)  = 0.075 µS, and (C)  = 0.06 µS. The white boxes constrain a region from 10 to 11°C where frequency is between .95 and 1.05 Hz. The horizontal lines represent the vertical boundaries of the box and are plotted explicitly below each map to demonstrate the high temperature variability. Line 1 –  = 0.0696 µS; line 2 –  = 0.0645 µS; line 3 –  = 0.0639 µS; line 4 –  = 0.0563 µS; line 5 –  = 0.0587 µS; line 6 –  = 0.0486 µS. White mark corresponds to bifurcation coalescence point that defines the parameter region for Figure 8.

Figure 8. Example of random parameter choices producing similar output at low temperature and divergent output at high temperature.

Temperature dependence fixed at:  =  = 1.5,  = 1.6,  = 3. From a chosen reference parameter point,  = 0.063 µS;  = 0.06 µS;  = 0.1 µS, at 11°C, each maximal conductance (leak, inward and outward) is given a ±7.5% tolerance and 15 randomly generated curves are plotted across temperature. Variability in parameter space is shown in the 3D plot above. The frequency, amplitude and duty cycle of the 15 models are plotted simultaneously across the three graphs as a function of temperature.