Journal of Neurophysiology

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Noisy Inputs and the Induction of On–Off Switching Behavior in a Neuronal Pacemaker

David Paydarfar, Daniel B. Forger, John R. Clay


Neuronal oscillators can function as bistable toggle switches, flipping between quiescence and rhythmic firing in response to an input stimulus. In theory, such switching should be sensitive to small noisy inputs if the bistable states are in close proximity, which we test here using a perfused squid axon preparation. We find that small noisy stimulus currents induce a multitude of paths between two nearby stable states: repetitive firing and quiescence. The pattern of on–off switching of the pacemaker depends on the intensity, spectral properties, and phase angle of stimulus current fluctuations. Analysis by spike-triggered averaging of the stimulus currents near the transitions reveals that sinusoidal stimuli timed antiphase or in phase with repetitive firing correlates with switching of the pacemaker off or on, respectively. Our results reveal a distinct form of bistability in which noise can either silence pacemaker activity, trigger repetitive firing, or induce sporadic burst patterns similar to those recorded in a variety of normal and pathological neurons.


Neurons with intrinsic oscillations in electrical activity regulate the functional states of diverse circuits (Llinás 1988) underlying motor control (Marder and Calabrese 1996), sensory processing (Derjean et al. 2003), behavioral state (Steriade et al. 1993), and memory formation (Marder et al. 1993). The ability for synaptic inputs to initiate or abolish rhythmic spike activity is important for shaping and modifying normal neural patterns in these circuits. Abnormal neuronal oscillations are implicated in certain disease states, such as repetitive firing of injured axons evoking painful paresthesia (Ochoa and Torebjörk 1980) or fasciculations (Bostock and Bergmans 1994) and rhythmic discharges of cortical neurons in patients with epilepsy (Avoli et al. 2005).

Here we are concerned with an interesting class of neuronal oscillators in which two stable states coexist for the same set of biophysical parameters. For such bistable cellular pacemakers, a rapid and persistent change in postsynaptic activity—switching the pacemaker on or off—is induced by a brief synaptic input. In this way, a stable switch in neuronal activity can occur without the need for a persistent change in the concentration of extracellular ions or neuromodulators or in signal transduction elements associated with adaptive plasticity.

In most studied cases of bistability, the two stable states have widely separated membrane potentials (Derjean et al. 2003; Kiehn 1991; Llinás and Sugimori 1980; Loewenstein et al. 2005; Marder et al. 1996). The membrane is hyperpolarized in the stable quiescent state. A brief and sufficiently large depolarizing stimulus induces a persistent depolarization of the membrane, called a “plateau potential.” Repetitive spikes or bursts of spikes are exhibited during this persistent depolarization, which can be switched off by a brief and sufficiently large hyperpolarizing stimulus that restores the membrane to the stable hyperpolarized quiescent state. These two states—hyperpolarized quiescence and a depolarized plateau potential with rhythmic firing—are separated by 10–50 mV, enabling each state to be robust in the face of small perturbations. These bistable neurons function as a “toggle switch,” flipping to the alternate state only in response to a sufficiently large stimulus.

Much less studied is bistability in which the stable states are in close proximity to each other (Canavier et al. 1993; Toerell 1971; Winfree 1980). In the simplest case, Huxley (1959) and others (Best 1979; Cooley et al. 1965; Rinzel 1978) showed that a small persistent inward current in the Hodgkin–Huxley equations (Hodgkin and Huxley 1952) creates a peculiar coexistence and close proximity of two stable states, one repetitively firing and the other quiescence at a stable equilibrium potential. Phase-plane analysis shows that the stable equilibrium potential is a fixed-point attractor nested within and in close proximity to a stable limit cycle attractor that represents repetitive firing. Experimental demonstration of the existence of highly proximate bistable states has been technically difficult, however, and implicated in only a few preparations in which a single pulse-shock stimulus must have an amplitude within a very narrow range and must be applied at a precise phase to switch the cell from stable repetitive firing to stable quiescence (Guttman et al. 1980; Jalife and Antzelevitch 1979; Shrier et al. 1990).

In theory, a pacemaker with coexisting and highly proximate bistable states should be highly sensitive to noisy perturbations of membrane potential (Forger and Paydarfar 2004; Paydarfar and Buerkel 1995; Schneidman et al. 1998). Because stochastic stimulation contains waveforms having diverse shapes, timings, and amplitudes, noisy perturbations of the oscillating potential can be used to find a large array of critical stimuli that induce a switch from repetitive firing to quiescence. Therefore noisy inputs could reveal a much greater accessibility to the quiescent state than has been appreciated from phase-resetting experiments that use a single pulse-shock stimulus protocol. In the present study we analyze the dynamics of noise-induced on–off switching behavior in the bistable squid giant axon, in which repetitive firing coexists with a nearby stable resting potential. Our results reveal a distinct form of bistability in which noise can either silence pacemaker activity, trigger repetitive firing, or induce sporadic burst patterns similar to those recorded in a variety of normal and pathological neurons.


Giant axon preparation

Experiments were performed on giant axons from the North Atlantic squid (Loligo pealei) at the Marine Biological Laboratory in Woods Hole, MA using axial wire voltage- and current-clamp techniques with intracellular perfusion as previously described (Clay and Shlesinger 1983). The external solution was artificial seawater, which consisted (in mM) of 430 NaCl, 10 KCl, 10 CaCl2, 50 MgCl2, and 10 Tris-HCl (pH = 7.5). The intracellular perfusate consisted (in mM) of 400 sucrose, 200 KF, 30 Na-glutamate, and 50 K-glutamate with the pH titrated to 8.5 using free glutamic acid. The temperature was in the 13–23°C range. In any given experiment it was maintained constant to within 0.1 by a negative feedback circuit connected to a Peltier device located within the experimental chamber. Bistability (stable firing coexisting with a stable resting potential) was previously shown to occur for these conditions (Clay and Shrier 2002).

Demonstration of membrane bistability

The axon preparation described above either rested at a stable membrane potential (see Table 1) or fired repetitively. Bistability was demonstrated using both current- and voltage-clamp circuitry. In the first case (resting state) a suprathreshold depolarizing current pulse of brief duration (1-ms duration) was applied to elicit repetitive firing. The membrane potential was then clamped at the original resting state. On release of the clamp, the axon was once again in the resting state. In the second case (repetitive firing) the axon was clamped at −60 mV (holding potential), which was adjusted in the hyperpolarizing or depolarizing potential until the net current recorded by the voltage clamp was 0. Bistable axons remained at that potential after release of the clamp. Sustained repetitive firing was reestablished using a brief duration current pulse. No external biasing current was given to the axon to achieve bistability.

View this table:

Characteristics of eight bistable axons

Stochastic stimulation

Stochastically varying current without offset was administered to the axon for 10-s periods using stimulus profiles generated by a computer program (MatLab, The MathWorks, Natick, MA) of a simple model of stochastically summated polysynaptic currents (PSCs) as further described below in Mathematical modeling. Excitatory PSCs and inhibitory PSCs were generated independently, each with a Poisson rate with a mean of 10 events per millisecond. Each PSC had an exponential rise time constant of 0.25 ms and decay time constant of 1 ms. These parameters were used in all experimental trials except as otherwise noted in results. The stimulus profile was the sum at any moment of all PSCs and the overall intensity of the stimulus was varied by changing the PSC amplitude. The computed stimulus profiles were converted to an analog stimulus using a digital-analog converter (National Instruments, Austin, TX) controlled by software (LabVIEW 6, National Instruments). The mean current over any single run was zero because the excitatory and inhibitory postsynaptic currents (EPSCs and IPSCs, respectively) had identical profiles and Poisson distributions. Intensity of stimulation is reported as the root-mean-square of the stimulus current (Irms) over the 10-s run.

Data acquisition and analysis

The transmembrane voltage and stimulus current were displayed and digitized (Windaq, DATAQ Instruments, Akron, OH) at 125 kHz per channel for storage on optical media and subsequent playback. Customized programs (C++, MatLab) were used for further analysis. The interspike interval (ISI) time series and ISI frequency histograms were determined for each run. A quiescent period was defined as subthreshold activity lasting for ≥30 ms. Repetitive firing was defined as two or more action potentials (APs) within a 30-ms period. Based on these definitions, we identified all transitions from repetitive firing to quiescence (RF→Q) and all transitions from quiescence to repetitive firing (Q→RF). Phase plots were created for both types of transitions by plotting the membrane current versus membrane voltage. Membrane current was determined by multiplying the membrane capacitance (1 μF/cm2) and the first derivative of membrane voltage (Guttman et al. 1980). Because the voltage data were acquired digitally, computation of its first derivative was achieved by creating a smooth Vm(t) function from the least-square fit of a third-order polynomial through 25 consecutive values of Vm around each desired point. Stimulus-current trajectories associated with RF→Q or Q→RF transitions were computed using event(action potential)-triggered averaging. The mean stimulus current was considered significant if the mean deviated from zero by ≥2 SE (Bryant and Segundo 1976).

Mathematical modeling

We applied a noise-based method (Forger and Paydarfar 2004) to identify stimuli that induce a switch between the bistable states of the Hodgkin–Huxley model of the space-clamped squid giant axon (Hodgkin and Huxley 1952). The model equations are Math Math Math Math where V is the membrane potential relative to the resting potential, m is the kinetics of sodium activation, h is the kinetics of sodium inactivation, and n is potassium activation. α and β are voltage-dependent terms that must be scaled to temperature by a Q10 of 3. Cm is membrane capacitance and g represents the conductances to sodium and potassium ions, as well as a nonspecific leakage conductance. We used the parameter values found in Table 3 of Hodgkin and Huxley (1952). The differential equations were integrated numerically using a variable-step fourth-order Runge–Kutta method. In the model equations, repetitive firing and quiescence can coexist if Ip is within a given range that depends on temperature, as detailed by Rinzel (1978).

Noisy stimulation Istim was computed from a Poisson distribution of IPSCs and EPSCs, defined as ±(1 − e−λ1t)e−λ2t (Forger and Paydarfar 2004). We simulated fast synapses with PSC rise time constant λ1 = 4 ms−1 and PSC decay time constant λ2 = 1 ms−1; inhibitory and excitatory PSCs were timed independently and randomly with a Poisson rate so that on average one excitatory and one inhibitory PSC occurred every 0.1 ms. Istim was the sum at any time of all IPSPs and EPSPs. The intensity of stimulation was defined as the root-mean-square of the applied current Math The bistable Hodgkin–Huxley equations were numerically simulated with Istim at intensities high enough to induce sporadic switching between RF and Q, but low enough so that the average rate of switching was small compared with the period of RF. We created a library of all stimuli around the time of RF→Q and of Q→RF transitions. The time of transition was defined as the AP peak closest to Q and we analyzed stimulus current within ±30 ms of the transitions. For a sufficiently large number of transitions, we found that the mean of all stimuli around each of the two types of transitions was significantly different from no stimulation (Bryant and Segundo 1976). The threshold intensity was determined by reducing the amplitude of the mean stimulus to the lowest level that caused a switch when applied back to the bistable Hodgkin–Huxley model.


Stable coexistence of two states: repetitive firing and quiescence

Membrane bistability was demonstrated in eight alkalinized axons (Table 1). The extracellular bath temperature (range 14.5–22.1°C) was well correlated (R = −0.87) with the baseline interspike interval (ISI) as previously reported by Clay and Shrier (2002). Figure 1A is an example of bistable behavior in one axon, in which the stable resting membrane potential (Vm) was −61 mV. A single current pulse (1 ms, 2.5 μA/cm2) induced stable repetitive firing with an ISI of 10.8 ms (Fig. 1A). A plot of membrane current Im versus Vm (Fig. 1, B and C) depicts the states of the membrane in the phase plane. Repetitive firing corresponds to a stable limit cycle (Fig. 1B). The coexisting stable rest potential (Vm = −61 mV, Im = 0) is near the limit cycle adjacent to its slowly depolarizing subthreshold trajectory (Fig. 1C). All eight axons had a similar configuration and close proximity of the coexisting resting and repetitive firing states.

FIG. 1.

Stable rest state and stable repetitive firing (RF) coexist. A: stable rest potential (−61 mV) switched to RF (period 10.8 ms) by current pulse (1 ms, 2.50 μA/cm2). B: RF corresponds to a clockwise limit cycle trajectory in the phase plane. C: phase plane with expanded scales relative to B reveals the remarkably close proximity of the stable rest state to the slowly depolarizing phase of the limit cycle. Arrows on the trajectories indicate direction of time.

Once the existence of bistability was established in each axon, we proceeded to administer stochastically varying currents (without offset) for periods of 10 s. Raising the intensity of stimulation (reported as the root-mean-square of the current, Irms; see methods) above a certain threshold caused switching from repetitive firing to quiescence (RF→Q) or quiescence to repetitive firing (Q→RF). Stimulation below the threshold for switching caused small fluctuations near the rest potential (±1 mV SD) if the axon was initially in the rest state. If the axon was repetitively firing, such subthreshold stimulation caused small (±0.2 ms SD) changes in the ISI from one cycle to the next.

Annihilation of repetitive firing by stochastic stimulation

We analyzed the remarkably small displacements of membrane potential and current associated with RF→Q transitions. The threshold of stimulus intensity for inducing an RF→Q transition varied between axons (mean Irms = 0.17 μA/cm2, range 0.05–0.22), but was consistent within an axon for a specified stochastic stimulus profile. Figure 2 shows an example in which the unstimulated axon exhibits stable repetitive firing (Fig. 2A). During stimulation (Irms = 0.13 μA/cm2) repetitive firing continued for four cycles, then the axon switched to subthreshold fluctuations of membrane potential (57 ± 1 mV SD) near the rest potential (Fig. 2, B and C), which persisted for the remainder of the 10-s period of stimulation. In the phase-plane limit cycles were seen before stimulation (Fig. 2A) and during the initial period of stimulation (Fig. 2B). However, after the fourth cycle after onset of stimulation, while the trajectory was slowly depolarizing there ensued a spiral-like collapse off the limit cycle and toward the rest state (Fig. 2B), followed by small clockwise oscillations near the rest potential that failed to restore the membrane back to the limit cycle (Fig. 2C). The results shown in Fig. 2 were found in all eight axons in which we observed a total of 28 instances of complete annihilation of repetitive firing by low-intensity stimulation. In all cases the membrane remained quiescent at the rest potential after cessation of the 10-s stimulus. The mean latency from onset of stimulation to the collapse of the limit cycle was 80 ms (range, 5–330 ms) using stimulation with Poisson time with a mean of 0.1 ms and PSC decay time constant of 1 ms. The latency was dependent in part on the Poisson time between PSCs. For example, in one trial (axon #6), stimulation with a Poisson time with a mean of 0.1 ms caused annihilation of repetitive firing 34 ms after onset of stimulation. In the second trial, we increased the Poisson mean time to 10 ms, which caused annihilation 1.7 s after onset of stimulation. The stimulus intensity was unchanged in the two trials (Irms = 0.06 μA/cm2). In another axon (#8), we noted that prolongation of the PSC time constant (without changing the stimulus intensity or the Poisson time) tended to increase in latency to switch. For example, for a PSC decay time constant of 1 ms the mean latency to RF→Q was 40 ms (three trials), whereas using a PSC decay time constant of 10 ms the mean latency to RF→Q was 128 ms (three trials). In both cases Irms = 0.18 μA/cm2.

FIG. 2.

Noisy inputs annihilate RF. A: stable RF in the absence of noise input. Stochastic stimulation [root-mean-square of the stimulus current (Irms) = 0.13 μA/cm2] was applied throughout the remainder of the record. B: RF was annihilated about 50 ms after application of noise. C: oscillations around the rest state. Brackets mark the segments plotted in the phase plane AC.

Sporadic burst patterns: dependency on the intensity of stimulation

The threshold of stimulus intensity for inducing a Q→RF transition varied between axons (mean Irms = 0.32 μA/cm2, range 0.22–0.44), but was consistent within an axon for a specified stochastic stimulus profile. For each axon, induction of a Q→RF transition required stochastic stimulation that was more intense (by a mean Irms of 0.15 μA/cm2) compared with the threshold stimulus that caused annihilation of repetitive firing. Consequently, raising the intensity of stimulation above the Q→RF threshold caused sporadic switching between periods of quiescence and periods of repetitive firing. Figure 3 provides an example in which the membrane is initially quiescent at the steady-state potential (−57 mV). We gave the same stimulus profile to the same axon as shown in Fig. 2, but with increased intensity Irms = 0.22 μA/cm2. Stimulation induced subthreshold fluctuations in membrane potential (Fig. 3A) followed by a burst of repetitive firing (Fig. 3B) and then the membrane switched back to subthreshold fluctuations (Fig. 3C).

FIG. 3.

Noisy inputs induce a burst of RF. Stimulation (Irms = 0.22 μA/cm2) induced subthreshold fluctuations followed by a burst of RF. Phase-plane plots: A: transition from rest state (a) to subthreshold fluctuations (b). B: transition from subthreshold fluctuations (a) to RF (b, c, d). C: transition from limit cycle (a, b) back to subthreshold fluctuations (c).

Sporadic switching between Q and RF was more frequent with more intense stimulation. Figure 4 shows an example in which the same time sequence of noise was administered in all runs but with increasing intensity. At the lowest intensity (Fig. 4A) the membrane was quiescent. Progressive increases in the stimulus intensity caused more frequent switching between the Q and RF states (Fig. 4, BD). When stimulation ended, an axon's subsequent behavior was determined by its state at the time of stimulus cessation. If the axon was in the Q state at the time of stimulus cessation (Fig. 5A), the membrane returned to the stable rest state. On the other hand, if the stimulus ended during a burst of RF (Fig. 5B), the membrane continued to exhibit RF. In both runs the stimulus intensity was 0.42 μA/cm2. These results corroborate the robustness of bistability throughout and after the 10-s period stimulation.

FIG. 4.

Switching behavior is more frequent with more intense stimulation. Stimulus was applied with increasing intensity from runs AD, but all runs have the same time sequence of noise. A: for Irms = 0.13 μA/cm2 the membrane is quiescent throughout stimulation. Note that this is the same stimulus that annihilated repetitive activity (Fig. 2). Progressive increases in the stimulus intensity (using the same stimulus profile) caused more frequent switching between quiescence and RF. B: Irms = 0.22 μA/cm2. C: Irms = 0.26 μA/cm2. D: Irms = 0.31 μA/cm2.

FIG. 5.

Membrane state at the time of stimulus cessation determines subsequent behavior. A: stable rest state persisted when axon was in the quiescent state at the time of stimulus cessation. B: stable RF persisted when stimulus ended during a burst of RF. Irms = 0.42 μA/cm2 in both A and B.

We analyzed the patterns of irregular bursting of action potentials (e.g., Fig. 4) by computing the ISIs during stimulation. Figure 6 shows plots of the results for one stimulation trial lasting 10 s (axon #4). The intrinsic period of repetitive firing of the axon in the unstimulated condition was 12.02 ± 0.05 (SD) ms. As shown in Fig. 6, A and B most of the cycle periods were near the intrinsic period of repetitive firing. There was a second preferred ISI centered at 24 ms. This second smaller peak in the histogram arises when a single cycle of subthreshold oscillation interrupts a train of action potentials. The shape of the ISI histogram depended on the intensity of stimulation, as shown in Fig. 7 with normalized histograms (AD) for various intensities in this axon. The plots are a stacked series of ISI histograms; the panels that are stacked farther away are ISI distributions resulting from more intense stimulation. For each panel, the ISI is normalized to the period of repetitive firing in the unstimulated condition and the frequency of observations is normalized as the percentage of the maximum number of observations. The plots show that the peak (mode) of each distribution was close to the period of the unstimulated condition, irrespective of the intensity of stimulation; however, the distribution broadens around this peak as the intensity of stimulation increases. There is a distinct smaller peak in the ISI distribution that centers around twice the period of the unstimulated condition. This second peak becomes more prominent as the stimulus intensity increases (Fig. 7, AC) then disappears in the histogram for the highest stimulus intensity (Fig. 7D).

FIG. 6.

Prolongations of interspike intervals (ISIs) are not random. Data are for one stimulation trial lasting 10 s (axon #4). Intrinsic period of repetitive firing of the axon in the unstimulated condition was 12.02 ± 0.05 (SD) ms. A: time series of ISI throughout the trial. B: ISI histogram showing that most cycle periods were near the intrinsic period of RF. There was a second preferred ISI around 24 ms.

FIG. 7.

Shape of the ISI frequency histogram depends on stimulus intensity. For each panel, the ISI is normalized to the period of repetitive firing in the unstimulated condition and the frequency of observations is normalized as the percentage of the maximum number of observations. A: Irms = 0.13 μA/cm2, n = 1,125 ISIs. B: Irms = 0.20 μA/cm2, n = 4,985 ISIs. C: Irms = 0.35 μA/cm2, n = 1,904 ISIs. D: Irms = 0.43 μA/cm2, n = 3,872 ISIs. Peak (mode) of each distribution was close to the period of the unstimulated condition, irrespective of the intensity of stimulation. Note that there was a smaller peak in the ISI distribution centered at twice the period of the unstimulated condition. This second peak becomes more prominent as the stimulus intensity increases (AC) then disappears in the histogram for the highest stimulus intensity (D).

Analysis of stimulus currents that drive the transitions in the membrane state

We found that stimuli with specific temporal profiles were associated with transitions between repetitive firing (RF) and quiescence (Q). This is revealed by superimposing and averaging stimuli associated with RF→Q transitions (Fig. 8A, with 21 transitions) and Q→RF transitions (Fig. 8C, with 22 transitions) during a 10-s trial. The tracings are aligned relative to the peak of the action potential at the transition (vertical dotted lines). The average Im versus Vm trajectories for all transitions in the trial are shown in the phase plane for RF→Q (Fig. 8B) and for Q→RF (Fig. 8D). These show the spiral paths that are characteristic of transitions between the limit cycle and the rest state.

FIG. 8.

Sinusoidal stimuli are associated with switching. Tracings are from a single 10-s trial during which there were 21 repetitive firing to quiescence (RF→Q) transitions (A and B) and 22 quiescence to repetitive firing (Q→RF) transitions (C and D). A and C: superimposed membrane potential recordings (top traces), stimulus current recordings (middle traces), and average stimulus current (bottom trace). All traces are aligned relative to the peak of the action potential at the transitions (vertical dotted lines). B and D: average trajectories in the phase plane for the RF→Q transition (B) and the Q→RF transition (D). Labels (a)–(e) show the timing of depolarizing and hyperpolarizing average stimulus and the corresponding trajectories in the phase plane. Dashed lines depict 2 SE above and below the average stimulus current.

In this trial, the period of repetitive firing in the unstimulated condition was 11.85 ± 0.04 (SD) ms. For the RF→Q transitions in the trial, the average stimulus current was a sinusoidal cycle with a period of 11.6 ms. The depolarizing wavelet is associated with the initial displacement off the limit cycle (Fig. 8, A and B, trajectory a), followed by a hyperpolarizing wavelet that is associated with the spiral back toward the rest potential (Fig. 8, C and D, trajectory b). For the Q→RF transitions, the average stimulus was also a sinusoidal cycle, with a period of 10.75 ms. The alternating depolarizing and hyperpolarizing wavelets were associated with a displacement away from the rest potential with progressively enlarging spiral trajectories that approach the limit cycle (Fig. 8, C and D, trajectories ae).

A consistent finding in all axons was that the mean stimulus associated with RF→Q transitions was a sinsusoid in antiphase with the membrane potential, whereas the mean stimulus associated with Q→RF transitions was a sinusoid nearly in phase with the membrane potential. These relationships in the timing of the stimulus wavelets relative to the membrane potential changes are shown in Fig. 9. The RF→Q transition (Fig. 9A, top) is defined by an action potential followed by damped sinusoidal cycles, shown as the average membrane potential of RF→Q transitions in a single 10-s trial (i.e., an enlargement of the transition in Fig. 8A). The corresponding average stimulus profile is a sinusoid that is nearly 180° out of phase with the damped oscillation in membrane potential. Figure 9A (bottom) shows the mean stimulus of all RF→Q transitions in this axon (511 transitions, 23 trails). Stimulus time for the pooled trials is normalized, with 1.0 representing the cycle period of repetitive firing in the absence of stimulation.

FIG. 9.

Timing of sinusoidal stimuli is critical for switching. A: RF→Q depicts the antiphase relationship between average membrane potential (top) and average stimulus current for a single trial with 21 transitions (middle) and for all 23 trials with a total of 511 transitions in this axon (bottom). Note that the stimulus is about 180° out of phase with the membrane potential. B: Q→RF depicts the in-phase relationship between average membrane potential (top) and average stimulus current for a single trial with 22 transitions (middle) and for all 23 trials with a total of 513 transitions (bottom). Stimulus time for pooled trials is normalized, with 1.0 representing one cycle period for the unstimulated condition. Dashed lines depict 2 SE above and below the average stimuli.

The Q→RF transition is shown Fig. 9B (top trace), with the corresponding average stimulus in a single trial (middle trace) and for all trials in the same axon (bottom trace, 513 transitions in 23 trials). The average stimulus associated with Q→RF transitions is a sinusoid that is roughly in phase with the membrane potential (lagging membrane potential changes by about 1 ms). For the pooled data in this axon, there was usually only one significant cycle of stimulation preceding the transition, but in some trials there were two or more cycles, such as the example shown in the middle trace of Fig. 9B and in Fig. 8, C and D.

Analysis of transitions in all eight bistable axons showed stimulus profiles similar to Figs. 8 and 9. The RF→Q transitions were associated with sinusoidal stimuli that were antiphase to the membrane potential and the Q→RF transitions were associated with sinusoidal stimuli that were in phase with membrane potential. The cycle period of the stimuli that induced the transitions were well correlated (R = 0.82) with the cycle period of repetitive firing (see Table 1) in the unstimulated condition.

Analysis of on–off switching behavior in the Hodgkin–Huxley model

The Hodgkin–Huxley (H-H) equations under space-clamped conditions provide a model for our experimental results (see methods). Bistability occurs over a range of persistent inward current and noisy current stimulation induces switching between repetitive firing and quiescence (Fig. 10A). In the H-H model the absolute noise intensities that cause annihilation of rhythmic firing and sporadic bursting patterns (similar to Fig. 4) depend on the temperature and on the strength of the persistent inward current, which has not yet been quantified in alkalinized axons. Nevertheless all experimental results (Figs. 19) are qualitatively replicated by simulation of the H-H equations. Using the noise-based search method described previously (Forger and Paydarfar 2004; also see methods) we computed the threshold stimulus that induces switching with the least amount of current. The stimulus is a sinusoid, timed antiphase to annihilate repetitive firing (Fig. 10B) and timed roughly in phase to induce repetitive firing (Fig. 10C). These stimulus shapes and timings are found for the full range of temperatures and persistent inward currents that result in bistability of the H-Huxley equations. It is noteworthy that a pulse stimulus can also induce the same transitions but much greater current amplitudes are required (Fig. 10, B and C).

FIG. 10.

Switching states in the bistable Hodgkin–Huxley equations. A: example of noise-induced switching. B and C: sinusoidal current stimulations are the smallest that induce RF→Q and Q→RF transitions, computed from a library of 435 stimuli (inducing 218 RF→Q transitions and 217 Q→RF transitions). Pulse stimuli (1 ms) can also induce switching but much higher amplitudes are required. Period of RF is 16.5 ms, T is 6.3°C, and Ip is 7 μA/cm2.

An important distinction between the H-H model and the preliminary model for excitability in the alkalinized squid giant axon (Clay and Shrier 2002) is that the range of depolarizing currents that produce bistability for the H-H model is narrow (Best 1979; Rinzel 1978), whereas bistability in the alkalinized preparation is exhibited over a larger range of parameters of the model. Although the exact boundaries for bistability in the Clay and Shrier (2002) model have not yet been delineated, the ionic basis for the robustness in bistability appears to be a combination of overestimation of gNa in the H-H model (Clay 2005) and a reduction by alkalinization of the background leak conductance, thereby unmasking INaP and increasing excitability of the membrane to produce bistability (Clay and Shrier 2002). In both models, bistable states are in close proximity along the voltage axis, in that because increases in depolarizing currents cause progressive separation of the fixed point from the limit cycle, the fixed point becomes unstable and the neuron changes over to a monostable pacemaker (Rinzel 1978).


Our study reveals novel properties of a neuronal pacemaker whose stable oscillatory state is in close proximity with a stable resting state, resulting in highly nonlinear responses of the cell to noisy input. Low noise levels fail to excite the cell in the resting state, yet rapidly annihilate repetitive firing, with the cell remaining quiescent thereafter except for small subthreshold fluctuations. Slightly more intense stimulation causes the cell to exhibit bursts of action potentials that are interrupted by periods of quiescence. The interspike intervals during bursts are highly regular and near the natural period of repetitive firing (Figs. 3 and 4). The pauses between bursts reflect switching to and from the resting state. The durations of these pauses are not random; over a range of moderate stimulus intensities there is a preponderance of pauses that are double the period of repetitive firing. The maximum duration of the pause is progressively reduced as the intensity of stimulation is progressively increased (Fig. 4). Removal of the noisy stimulus causes no further switching behavior and the cell remains in the same state at which it happened to be in at the time of stimulus cessation (Fig. 5).

The geometrical description of switching in the phase plane suggests that sinusoidal stimuli initiate repetitive firing by growth of amplitude of the cycle around a focus (the resting state), whereas sinusoidal stimuli terminate repetitive firing by decreasing the cycle amplitude around the focus. This geometry of switching behavior can be inferred from known biophysical mechanisms of excitability. In the quiescent state, membrane potential exhibits subthreshold oscillations in response to small perturbing stimuli. These oscillations are maximally induced when the stimulus fluctuates with a frequency at the natural resonance frequency of the membrane (Gutfreund et al. 1995; Izhikevich et al. 2003). The biophysical mechanism is a form of phenomenological impedance, arising directly from the nonlinear voltage-dependent and time-variant conductances of the fast sodium and potassium currents (Mauro et al. 1970). We believe that stimulus-evoked resonance of membrane potential induces a switch from quiescence to repetitive firing. The key evidence is that the stimulus current exhibits an oscillation that is nearly in phase with the oscillations in membrane potential just before the onset of repetitive firing (Fig. 9B). A related finding revealed by ISI histograms is that the likelihood of switching the pacemaker “on” is highest when the quiescent period is approximately two cycle periods. This can occur when modest noisy stimulation induces subthreshold oscillations near the period of repetitive firing. Subthreshold oscillations driven by noise are not highly regular, which might explain the relatively small ISI peak at twice the subthreshold period and the absence of additional peaks at higher-integer multiples of the subthreshold period that are expected in neurons with highly regular subthreshold oscillations (Longtin et al. 1991). The relatively narrow range of ISIs during repetitive firing is consistent with the observation that under the conditions of our studies the squid giant axon exhibits type 2 excitability, characterized by a narrow-frequency range of repetitive firing, in contrast to the broad-frequency range of neurons with type 1 excitability (Hodgkin 1948; Rinzel and Ermentrout 1989; Tateno et al. 2004). These observations taken together support the view that induction of repetitive firing by noisy input to the cell arises from resonance at the natural frequency of the membrane and the timing of the switch is associated with sinusoidal current stimuli that are in phase with subthreshold oscillations in membrane potential.

The switch from repetitive firing to quiescence is also associated with sinusoidal stimulation but, in contrast to the resonance mechanism, we find that the sinusoid is nearly 180° out of phase with the pacemaker potential. The antiphase stimulus thus induces a depolarizing effect while the membrane is hyperpolarized and a hyperpolarizing effect while the membrane is depolarized. In the phase plane the effect of antiphase sinusoidal stimulation has the appearance of “peeling” the membrane trajectory away from the stable limit cycle toward the stable resting state. It is important to note, however, that our protocol using noisy input generates a large array of stimulus shapes, timings, and intensities—each capable of extinguishing the pacemaker potential. The sinusoidal shape is the mean of this array of stimuli (Agüera y Arcas et al. 2003; Forger and Paydarfar 2004) and the spiral path from the limit cycle to the resting state is the mean of the corresponding membrane state trajectories. We find in our numerical simulations of the Hodgkin–Huxley equations that sinusoidal stimuli are capable of inducing switching between repetitive firing and quiescence with currents that are lower than pulse stimuli (Fig. 10). This was previously demonstrated in the Fitzhugh–Nagumo (F-N) model of excitability in which calculus of variations also reveals that sinusoidal stimuli initiated or extinguished neuronal firing with the least amount of current (Forger and Paydarfar 2004). The period and timing of these optimal sinusoidal functions with respect to repetitive firing of the F-N model is similar to those found in our experiments.

Multisynaptic input and membrane channel noise are ubiquitous and induce stochastic fluctuations in membrane potential (Traynelis and Jaramillo 1980). The manner in which neurons process information that is inherently noisy is of considerable interest (Moss et al. 2004), with a large body of experimental and modeling work on spike timing that assumes each neuron is intrinsically quiescent and monostable. There has been no definitive evidence for the existence of noise-sensitive bistable neurons in normal circuits. Nevertheless, the sporadic burst patterns in our preparation are reminiscent of the firing patterns observed in a variety of preparations such as mammalian neocortical neurons (Llinás et al. 1991), some types of sensory receptors (Braun et al. 1994), and neurons within the suprachiasmatic nucleus that regulate circadian rhythms (Kononenko et al. 2004). We speculate that pacemakers with the properties described in our study may be common yet underrecognized because of technical difficulties in accessing the highly proximate resting state using a shock-stimulus protocol. Externally applied noisy perturbations of the neuron can be used as an experimental probe for the identification and analysis of bistable neuronal pacemaker with highly proximate states.

If such bistable neurons exist in vivo, what might be the functional significance? This question is difficult to address given the paucity of relevant information, but one can speculate based on the striking nonlinearities in switching responses. For example, a population of such neurons that are coupled to each other could switch from quiescence to synchronized rhythmic firing in response to a tiny stimulus acting on only a small subgroup or even a single neuron. This mechanism for synchronization could be triggered with a very small energy of activation, sufficient to displace the neuron(s) from the resting state to the highly proximate limit cycle. Excitatory coupling to other quiescent bistable neurons would synchronize additional pacemakers in the circuit. Pacemakers whose phase drifts away from the aggregate phase would be switched to the quiescent state if the aggregate rhythm results in an antiphase signal acting on the stray neuron, a mechanism that would further enhance the coherence of the ensemble of cellular oscillators. Previous work highlighted the importance of resonance in generating and transmitting a burst of action potentials with a specified frequency across a network of neurons (see review by Izhikevich et al. 2003). The bistable neuron of our study exhibits the requisite properties for such behavior, with an additional attribute that bursts of action potentials on the timescale of seconds can be initiated or terminated by brief stimuli on the order of milliseconds. The dynamic-clamp methodology (Prinz et al. 2004) could provide insight into the critical changes in membrane conductances induced by synaptic inputs that are required for initiating a burst, synchronizing neighboring cells, and terminating burst activity.

Our findings may be relevant to pathological conditions that cause hyperexcitability and repetitive firing of cells. Injured or demyelinated axons exhibit spontaneous switching between repetitive firing and quiescence (Baker 2000; Kapoor et al. 1997) and the ectopic firing patterns result in painful paraesthesia (Ochoa and Torebjörk 1980) and fasciculations (Bostock and Bergmans 1994). Persistent depolarizing currents are implicated in producing the pathological hyperexcitability of the axons (Baker 2000). Inherited sodium channelopathies are another group of disorders in which small persistent sodium currents lead to pathological hyperexcitability, with specific mutations that cause skeletal muscle myotonia (Cannon et al. 1991), ventricular fibrillation (Bennett et al. 1995), and epileptic seizures (Lossin 2002). In these paroxysmal disorders spontaneous switching is observed between normal and pathological states and could reflect bistable membrane states that switch in response to small perturbing stimuli, such as modeled for myotonia by Cannon et al. (1993). Our experimental findings suggest that pathological repetitive firing arising from small persistent depolarizing currents might be rapidly silenced by low-level noisy inputs and that from the stochastic signal one can estimate a stimulus shape and timing for selectively extinguishing the pathological pacemaker activity.


This research was supported by the Frederik B. Bang, M.G.F. Fuortes, John O. Crane, and H. Keffer Hartline Fellowships of the Marine Biological Laboratory, Woods Hole, Massachusetts and by the Intramural Research Program of the National Institute of Neurological Disorders and Stroke.


We are grateful to J. Moore, W. Schwartz, M. Hines, and P. Grigg for discussions and comments on the manuscript and to D. Robichaud for assistance in data analysis and preparation of the figures.


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