Abstract
White, John A., Ruby Klink, Angel Alonso, and Alan R. Kay. Noise from voltagegated ion channels may influence neuronal dynamics in the entorhinal cortex. J. Neurophysiol. 80: 262–269, 1998. Neurons of the superficial medial entorhinal cortex (MEC), which deliver neocortical input to the hippocampus, exhibit intrinsic, subthreshold oscillations with slow dynamics. These intrinsic oscillations, driven by a persistent Na^{+} current and a slow outward current, may help to generate the theta rhythm, a slow rhythm that plays an important role in spatial and declarative learning. Here we show that the number of persistent Na^{+} channels underlying subthreshold oscillations is relatively small (<10^{4}) and use a physiologically based stochastic model to argue that the random behavior of these channels may contribute crucially to cellularlevel responses. In acutely isolated MEC neurons under voltage clamp, the mean and variance of the persistent Na^{+} current were used to estimate the single channel conductance and voltagedependent probability of opening. A hybrid stochasticdeterministic model was built by using voltageclamp descriptions of the persistent and fastinactivating Na^{+} conductances, along with the fast and slow K^{+} conductances. All voltagedependent conductances were represented with nonlinear ordinary differential equations, with the exception of the persistent Na^{+} conductance, which was represented as a population of stochastic ion channels. The model predicts that the probabilistic nature of Na^{+} channels increases the cell's repertoire of qualitative behaviors; although deterministic models at a particular point in parameter space can generate either subthreshold oscillations or phaselocked spikes (but rarely both), models with an appropriate level of channel noise can replicate physiological behavior by generating both patterns of electrical activity for a single set of parameters. Channel noise may contribute to higher order interspike interval statistics seen in vitro with DC current stimulation. Models with channel noise show evidence of spike clustering seen in brain slice experiments, although the effect is apparently not as prominent as seen in experimental results. Channel noise may contribute to cellular responses in vivo as well; the stochastic system has enhanced sensitivity to small periodic stimuli in a form of stochastic resonance that is novel (in that the relevant noise source is intrinsic and voltagedependent) and potentially physiologically relevant. Although based on a simple model that does not include all known membrane mechanisms of MEC stellate cells, these results nevertheless imply that the stochastic nature of small collections of molecules may have important effects at the cellular and network levels.
INTRODUCTION
Electrical noise plays an important, if poorly understood, role in the functioning of the nervous system. The principal source of this noise is assumed typically to be synaptic, resulting from the probabilistic release of synaptic vesicles. Contributions of the inherently stochastic nature of voltagegated ion channels to neuronal noise levels are widely assumed to be minimal because of the large number of channels involved. However, evidence from experimental (Johansson and Århem 1994; Sigworth 1980; Verveen 1961), theoretical (Chow and White 1996; Lecar and Nossal 1971a,b) and computational (Rubinstein 1995; Schneidman et al. 1998; Skaugen and Walløe 1979; Strassberg and DeFelice 1993) studies indicates that noise from voltagegated channels can have important effects at the cellular level.
The possible importance of intrinsic noise from voltagegated channels is emphasized by recent results demonstrating that additive noise can allow dramatic improvements in representations of weak signals in neuronal systems (Bezrukov and Vodyanoy 1995; Collins et al. 1996; Douglass et al. 1993; Levin and Miller 1996). This result holds for systems of one nonlinear unit and, more dramatically, for systems that sum the responses of parallel units (Collins et al. 1995). If stochastic ion channels are an important neuronal noise source, they may serve to boost responses to weak signals as well.
Here we study the effects of channel noise in neurons of the superficial medial entorhinal cortex (MEC), which are responsible for delivering information from the association cortices to the hippocampus via the perforant path. These neurons exhibit subthreshold oscillations in membrane potential at a frequency of ∼8 Hz (Alonso and Klink 1993; Alonso and Llinás 1989). The subthreshold oscillations are generated by a mechanism involving a persistent (noninactivating) Na^{+} conductance and a slow outward conductance (Klink and Alonso 1993; White et al. 1995). The persistent Na^{+} conductance is small (White et al. 1995) and, consequently, thought to be mediated by a modest number of ion channels.
The approach used here involves three steps. First, we characterize the population of persistent Na^{+} channels under voltage clamp in acutely isolated cells from the MEC. This characterization gives us a mathematical description of the properties of the channels. Second, we construct a computational model to represent the major populations of ion channels responsible for subthreshold oscillations and spiking behavior. In this model persistent Na^{+} channels are represented as a population of identical stochastic devices with two states (open and closed). Other, larger populations are represented in deterministic HodgkinHuxley form. We show that the range of qualitative behaviors that the model can exhibit depend crucially on the level of channel noise and tune the model to best describe currentclamp results from MEC neurons. Third, we use the tuned model to demonstrate that channel noise may make important contributions to interspike interval statistics and the probability of responding to a weak, physiologically relevant stimulus. Two specific properties of the channel noise—its voltagedependence and bandwidth—make important contributions to these results.
METHODS
Experimental methods
ISOLATED NERVE CELLS.
Isolated cells were studied by using previously published methods (White et al. 1993). Briefly, male rats (125–200 g) were anesthetized with pentobarbitol sodium (Nembutal; 30 mg/kg) and decapitated. The brain was quickly removed and a block of tissue containing the retrohippocampal region was cut. Horizontal slices (350 μm) were cut in cold (6–10°C), oxygenated Ringer solution containing (in mM) 115 NaCl, 5 KCl, 1 CaCl_{2}, 4 MgCl_{2}, 20 piperazineN,N′bis(2ethanesulfonic acid), and 25 glucose. Slices were cut into chunks containing only the superficial MEC. These chunks were transferred to a stir flask, brought to 30°C, reacted with proteinase K (0.2 mg/ml) for 5 min and trypsin (1 mg/ml) for 30 min, and allowed to return to room temperature. Dissociated cells were isolated from the MEC chunks by mechanical trituration. Recordings were made in a solution containing (in mM) 120 NaCl, 5 CsCl, 2 CaCl_{2}, 2 MgCl_{2}, 15 tetraethylammonium Cl, 5 4aminopyridine, 10 N2hydroxyethylpiperazineN′2ethanesulfonic acid (HEPES), and 25 glucose, pH = 7.4. Voltageclamp recordings were made by using the whole cell patch technique with an Axopatch 200A amplifier (Axon Instruments). The recording solution contained (in mM) 120 CsF, 15 NaCl, 10 HEPES, and 11 ethylene glycolbis(βaminoethyl ether)N,N,N′,N′tetraacetic acid (R = 1–3 MΩ). Data were lowpass filtered with a cutoff frequency of 5 kHz and sampled at 20 kHz.
BRAIN SLICES.
Horizontal brain slices of the MEC (350 μm) were cut by following previously published methods (Alonso and Klink 1993). Recordings were made in Ringer solution containing (in mM) 124 NaCl, 5 KCl, 1.2 KH_{2}PO_{4}, 2.4 CaCl_{2}, 2.6 MgSO_{4}, 26 NaHCO_{3}, and 10 glucose, pH = 7.4 while saturated with 95%O_{2}5% CO_{2}. The rather large value of extracellular K^{+} concentration [K^{+}]_{o} = 6.2 mM was used for continuity with previous work in this system (Alonso and Klink 1993; Alonso and Llinás 1989), but very similar results are obtained with lower concentrations; in fact, the experimental data from Fig. 4 were collected with [K^{+}]_{o} = 3 mM. Layer 2 of the MEC was identified via transillumination. Sharp electrodes, filled with potassium acetate (3 M; R = 40–80 MΩ), were used for intracellular currentclamp recordings, which were made with the use of an Axoclamp 2B amplifier (Axon Instruments). Data were lowpass filtered with a cutoff frequency of 1 kHz and sampled at 2.5 kHz.
Modeling methods
The singlecompartment model is described by the currentbalance equation: C_{m} dV_{m} /dt = −ḡ _{Na1} m ^{3} _{∞}(V _{m})h(V _{m},t)[V _{m} − V _{Na}] − ḡ _{K1} n ^{4}(V _{m},t)[ ̅V̅ ̅ ̅ ̅m̅ ̅ ̅ − V _{K}] − N _{o}γ[V _{m} − V _{Na}]/SA − ḡ _{K2} w(V _{m},t)[V _{m} − V _{K}] − ḡ _{L}[V _{m} − V _{L}] +i _{app}, where C _{m} = 1.5 μF/cm^{2}, ḡ _{Na1} = 53 mS/cm^{2}, ḡ _{K1} = 30 mS/cm^{2}, ḡ _{L} = 0.1 mS/cm^{2}, N _{o} = the number of open persistent Na^{+} channels, γ = the singlechannel conductance of the persistent Na^{+} channel, SA = cellular surface area, V _{Na} = 55 mV, V _{K} = −69 mV, and V _{L} = −54 mV. [The value of C _{m} used is slightly larger than recent results (Major et al. 1994) would indicate, but is compatible with estimates we have made in the past (White 1990) and with the notion that our rough estimates of surface area are low because they neglect the effects of enfolding. In any event, inaccuracies in the value of C _{m} can be offset by scaling conductance values in the currentbalance equation.] In tuning the model to match experimental data, ḡ _{K2} was varied between 8.5–13 mS/cm^{2}; ḡ _{Na2} = γp _{0 } N/SA (where p _{0} and N are the maximal probability of opening and number of persistent Na^{+} channels, respectively) was varied from 0.15–0.3125 mS/cm^{2}. Changes in N or ḡ _{Na2} were made independent of other parameters by altering surface area SA. By keeping all other aspects of the model the same, this method allowed us to examine the effects of these changes in a wellcontrolled manner. Each of the deterministic gating variables (m, h, n) is described by a firstorder, nonlinear ordinary differential equation (ODE) of HodgkinHuxley form. Mathematical descriptions of gating variables and other parameters were derived from voltageclamp data from isolated cells (Eder et al. 1991; White et al. 1993, 1995).
Stochastic persistent Na^{+} channels were modeled with the use of a previously published method (Chow and White 1996) that relies on the joint probability distribution of any state transition in a system of N identical, independent ion channels. We used a simple twostate Markov process model of the persistent Na^{+} channels not because we believe it to be correct—persistent Na^{+} currents undoubtedly arise via complex ion channel kinetics (e.g., Alzheimer et al. 1993)—but rather because it was the simplest model that fit the available voltageclamp data, matching the mean and coefficient of variation (the ratio of SD to mean) of the steadystate, tetrodotoxin (TTX)sensitive Na^{+} current within the relevant range of membrane potentials. An activation time constant τ = 0.15 ms was used for the persistent current to match the high (>1 kHz) bandwidth of this current (Fig. 1 C).
Data analysis
COUNTING CHANNELS.
The procedure for characterizing an assumed homogeneous population of ion channels is based on a binomial model in which the mean conductance g(V_{m} ) = Nγp(V_{m} ), where N is the number of channels in the population, γ is the singlechannel conductance, and p(V_{m} ) is the voltagedependent probability that each channel is open. Under this model the variance of conductance, also voltagedependent, is given by the equation σ^{2} _{g}(V _{m}) = Nγ^{2} p(V _{m})[1 − p(V _{m})]. p(V _{m}) can be parameterized by the function p(V _{m}) = p _{0} f(V _{m}), where p _{0} is the maximal probability of opening of the channel and f(V _{m}) = 1/[1 + exp(−(V _{m} − V _{0})/dV)] is a Boltzmann function. Under ideal conditions one would use these two relationships, along with measurements of g and σ_{g} over the entire range of relevant potentials, to estimate N, γ, and the form of p(V _{m}). In our case the diminutive size of the persistent Na^{+} conductance left us with reliable data only for potentials less than or equal to −30 mV. We estimated N, γ, p _{0}, V _{0}, and dV by using a multistep process. First, we divided mean and variance data by the value of g at −30 mV to allow comparison among different cells (Fig. 1 B). This normalization procedure is equivalent to multiplying each by a factor α = [Nγp _{0} f(−30)]^{−1}. Normalized mean and variance data were fit simultaneously by using the functions βγp _{0} f(V _{m}) and βγ^{2} p _{0} f(V _{m})[1 − p _{0} f(V _{m})], respectively, where β = αN. Values of the parameters γ and p _{0} were searched over discrete spaces with mesh size 5 pS and 0.05, respectively. Once the values of β, γ, p _{0}, V _{0}, and dV were determined from the lumped data, the estimated value of N for each cell was determined from the relationship n = β/α. Channel counts from isolated cells were taken only as preliminary estimates that were refined by fitting the bifurcation and powerspectral properties of currentclamp recordings from MEC cells in brain slices.
ANALYZING SPIKE CLUSTERS.
Clustering was analyzed in sequences of interspike intervals, calculated offline from digitized data. A cluster of M spikes was defined by the presence of M1 adjacent intervals of length 0.5–1.5 T, where T is the period of underlying subthreshold oscillations, determined from the fast Fourier transform (FFT) power spectrum. Isolated action potentials were considered clusters of length M = 1. The probability of observing a cluster of length M [p(M)] was estimated by dividing the number of observed clusters of length M by the total number of clusters in the data set.
RESULTS
We studied the persistent Na^{+} conductance under whole cell voltage clamp in dissociated MEC neurons. Figure 1 A, inset, shows voltageclamp responses (prepulse potential = −90 mV, pulse potential = −40 mV) in the presence (top trace) and absence (bottom trace) of 1 μM TTX. In the absence of TTX, the major membrane current is a large, fast Na^{+} current (clipped in this trace) that settles to a persistent value of approximately −70 pA. In TTX both the fast and persistent Na^{+} currents are blocked, leaving only a transient inward current (probably carried by Ca^{2+}, but not studied in detail).
The points in Fig. 1 A show the currentvoltage relationship for the persistent Na^{+} current, derived from voltageclamp responses by subtracting records in 1 μM TTX from control records and averaging over the last 200 ms of the subtracted record. The solid line shows a leastsquares fit of a Boltzmann functiondriving force product to the data.
For membrane potentials between −60 and −35 mV and with other conductances blocked, the Na^{+}channel mediated component is responsible for the great majority of the mean and as much as 80% of the variance in total recorded current (e.g., compare the means and variances of the top and bottom traces in Fig. 1 A, inset). The steadystate mean and variance of the persistent Na^{+} conductance are shown for several cells over a range of clamp potentials in Fig. 1 B. To allow comparisons among cells, both means and variances have been normalized by the mean value of the conductance at V _{m} = −30 mV (see Methods). Using these values and a binomial modeling method we estimated the number of channels N, singlechannel conductance γ, maximum probability of opening p _{0}, halfactivation point V _{0}, and steepness of the activation function dV. From this procedure we obtained the following estimates: γ = 20 pS, p _{0} = 0.125, V _{0} = −37.5 mV, and dV = 6.5 mV. The estimated value of N ranged from 388 to 2,498 with a mean value of 1,276 (n = 4). In this paper we take the values of γ, p _{0}, V _{0}, and dV as set, but recalibrate the value of N for cells in brain slices, which contain more of their processes and thus potentially more Na^{+} channels, by matching currentclamp data (see Fig. 3).
The frequency content of the TTXsensitive persistent Na^{+} current was examined by taking FFTs of subtracted traces. Figure 1 C shows the power spectrum from the last 200 ms of the difference between the current traces in Fig. 1 A, inset. As exemplified by this trace, power spectra were consistently flat (or in some cases, displayed possible 1/f ^{α} behavior) for frequencies <1–3 kHz. Bandwidth limitations of our recordings prevent us from knowing the cutoff frequency f _{0} of these records, but in all cases examined this cutoff was ≥1 kHz, implying that the activation time constant τ = 1/(2πf _{0}) of the persistent Na^{+} channels is ≤0.16 ms.
Because the population of persistent Na^{+} channels [or alternatively, the population of Na^{+} channels that are in the persistent “mode” (Alzheimer et al. 1993)] contributes significantly to total membrane noise, it stands to reason that the electrical noise induced by a flicker of the probabilistic channels may have an effect on macroscopically measured oscillations. To test this hypothesis we constructed a model of sub and suprathreshold behavior in MEC cells. The model consisted of a membrane capacitance in parallel with a leak conductance and four nonlinear conductances (see Methods). Two of the nonlinear conductances—the persistent Na^{+} current and a slow K^{+} current—generate subthreshold oscillations. The remaining two—a fast, inactivating Na^{+} current and a delayed rectifier—underlie action potentials. The model differs from typical models of excitable membranes in that the persistent Na^{+} conductance is modeled not with deterministic, nonlinear differential equations but rather as a set of twostate (open and closed) Markov processes. The parameters of this Markov model were chosen to mimic the most fundamental and consistent features of the persistent Na^{+} channels (specifically, the mean and variance of the steadystate currentvoltage relationship, as well as the broadband nature of the channel noise). The remaining three active conductances are 2–3 orders of magnitude larger than g _{Na2}, implying that they can be modeled sufficiently by using deterministic methods.
Figure 2 shows maps of this qualitative behavior as a function of the maximal magnitudes of the persistent Na^{+} conductance (ḡ _{Na2} = γp _{0 } N/SA) and slow potassium conductance (ḡ _{K2}) for the following three model types: deterministic, stochastic with n = 1,200, and stochastic with N = 4,800. The five shades, labeled as Regions 1–5 at the right, represent regions of qualitative behavior for stimulation over a large range of positive DC currents. From bottom right to top left, Region 1 represents models that are silent for all levels of current injection; Region 2, models that are quiet at rest and generate subthreshold oscillations but no spikes with positive DC current; Region 3, models that generate quiescence, subthreshold oscillations, and spikes, replicating results from intracellular recordings; Region 4, models that exhibit quiescence and spikes but no subthreshold oscillations; and Region 5, models that spike spontaneously.
The three model types show significant differences in qualitative behavior. The map for deterministic models is dominated by the quiescent (Region 1) and spiking (Regions 4 and 5) regions. The map for stochastic models with N = 1,200 is dominated by Regions 1, 2, and 5. This model type shows more excitability than the deterministic model, but again Region 3 is sparsely represented. The map for stochastic models with N = 4,800 shows all five regions. Most importantly, it is the only map of the three that reliably mimics the cells' ability to generate spikes, subthreshold oscillations, and phaselocked spikes without an exotic mechanism involving currentdependent modulation of maximal conductance values. We take this as evidence that N ≈ 4,800 in intact cells. The fact that this estimate is higher than estimates obtained from isolated cells may reflect the fact that in the isolation procedure, the cell loses most of even its most proximal processes and hence some fraction of its voltagegated ion channels.
We determined bestfit values of ḡ _{Na2}, ḡ _{K2}, and N for further analysis by searching systematically through this parameter space and matching power spectra at a given level of depolarization to those from experimental results. Figure 3, A–B, shows examples from this analysis with ḡ _{Na2} = 0.25 mS/cm^{2} and ḡ _{K2} = 10.6 mS/cm^{2}. Plotted are results with varying N in the time (A) and frequency (B) domains. As N increases, the simulated response becomes less energetic, approaching the quiescent behavior of the related deterministic system at this level of depolarization. The N = 4,800 case matches the experimental results for this and higher (Fig. 3, C–D) levels of depolarization. Experimental and simulated results match well in peak power, peak frequency, and highfrequency slope, differing mainly in that the experimental data exhibit a noise floor at high frequencies and more lowfrequency energy. Below, we use these values of ḡ _{Na2}, ḡ _{K2}, and N exclusively.
If depolarized sufficiently, MEC stellate cells and our stochastic model exhibit action potentials that rise from the depolarizing phase of the subthreshold oscillations. As exemplified by the traces of Fig. 4 A, these action potentials do not occur randomly but rather in distinct “spike clusters” (i.e., on ≥2 adjacent cycles). Spike clustering can be quantified by measuring the probability of observing a spike cluster of M spikes (M = 1, 2, 3,⋅⋅⋅). Figure 4 B shows a plot of p(M) versus M for experimental data (solid symbols) and simulations with channel noise (open symbols). Also shown is the expected value of p(M) for a “memoryless” process with the same probability of spiking per cycle (0.22) but with no dependence of spiking probability on past history. Models with channel noise show evidence of spike clustering, in the form of enhanced p(M) for M > 1. This effect does not depend critically on the bandwidth of the channel noise; increasing the activation time constant of the persistent Na^{+} channels to a value 5 times slower than we estimated above has only mild effects on clustering (open triangles; we believe the differences seen between the open circles and open triangles are due to necessary changes in mean V _{m} necessary to control for spiking probabilities in these 2 models). Spike clustering appears strongest in experimental data, for which clusters of length M = 2 are most probable, although limitations in the amount of data make this conclusion less than firm. If real, this preponderance of clusters of length M = 2 in the experimental results may be due to rebound spikes not accounted for in our model or other, more complex dynamical influences (Rush and Rinzel 1995; Wang 1993).
Under in vivo conditions in the theta state, cells of the MEC are driven at theta frequencies by their peers and respond in phase with the input stimulus. Results from Fig. 2 show that channel noise increases excitability, pointing to the possibility that MEC neurons may use channel noise to enhance their responses to weak periodic stimuli in a form of stochastic resonance. We examined this possibility by comparing responses of deterministic and stochastic models to periodic excitatory input. Under modeled conditions the deterministic model jumps from a purely subthreshold response (Fig. 5 A, leftmiddle trace) to a 3.5Hz doublet response (lefttop) over a narrow range of input conductance magnitudes (Fig. 5 B, solid line). A continued increase in the magnitude of the synaptic conductance drives the system to respond every cycle of the 7Hz input.
The stochastic model behaves quantitatively and qualitatively differently. Channel noise renders this model more sensitive to small stimuli (Fig. 5). For the control model (Fig. 5 A, right traces; Fig. 5 B, dashed line) with rapid persistent Na^{+} channels, the probability of a response per cycle remains higher than that of the deterministic model for nearly all values of g _{syn}, and responses are nearly always single spikes rather than doublets. (Stochastic models do show doublets reliably for nearby regions of parameter space.) In contrast to spikeclustering results with DC stimulation (Fig. 4), response probabilities to AC stimulation depend on the bandwidth of the channel noise. Slowing down the activation time constant of the persistent Na^{+} channels by a factor of 5 (Fig. 5 B, dotted lines) gives an enhanced probability of response only for small values of g _{syn}. This manipulation has no effect on the underlying deterministic system (data not shown), indicating that the difference between the dashed and dotted lines in Fig. 5 B is a property of channelnoise bandwidth rather than one of the related deterministic dynamics.
DISCUSSION
In our simulations two factors make channel noise an important contributor to cellular response properties in the MEC. First, a relatively small number of persistent Na^{+} channels gives a relatively high coefficient of variation (CV; the ratio of SD to mean is proportional to N ^{−1/2}) in induced current. Second, this conductance is strategically positioned (via its activation properties and associated reversal potential) so that its activation drives the cell from near resting potential to spike threshold. Together, these factors make many of the general effects of channel noise rather insensitive changes in the specific form of the noise; changing the threshold (data not shown), bandwidth (Figs. 45), or voltage dependence (data not shown) of the noise altered the details but not the basic properties of our results. This robustness lies in the simple fact that the amount of channel noise we measure directly (Fig. 1) is sufficient to deflect the trajectory of the voltage signal as it traces a path in phase space. Because the effects of channel noise are robust, we expect such effects to be significant in many nerve cells that exhibit persistent Na^{+} conductances (Llinás 1988), as well as in other neurons without prominent persistent Na^{+} currents (Chow and White 1996; Johansson and Århem 1994; Lecar and Nossal 1971a,b; Rubinstein 1995; Schneidman et al. 1998; Sigworth 1980; Skaugen and Walløe 1979; Strassberg and DeFelice 1993; Verveen 1961). Effects of the kind we describe may be important more generally, as exemplified by recent work implicating stochastic chemical reactions in other nonlinear physiological systems, including retinal transduction (Felber et al. 1996) and regulation of gene expression (McAdams and Arkin 1997).
Among the putative effects of channel noise reported here is a form of stochastic resonance (SR) with weak periodic stimuli. The noise source underlying this effect differs from that used in other neuronal studies (Collins et al. 1996; Douglass et al. 1993; Levin and Miller 1996) in two ways. First, the noise source in our case is an intrinsic characteristic of the system rather than experimentally imposed. Thus rather than relying on the input's having an appropriate level of noise to boost but not overwhelm pertinent weak inputs, the cell has the opportunity to modify its noise level dynamically. Second, the level of channel noise in MEC neurons is voltage dependent and thus covaries with the level of excitation. This voltage dependence has demonstrable effects on the model's behavior; although the stochastic simulation results of Fig. 4 can be replicated by a deterministic model with additive, voltageindependent noise, spike clustering at lower spiking probabilities occurs only with simulated channel noise and not with voltageindependent noise (data not shown). The voltage dependence of channel noise may contribute to the results of Fig. 5 as well.
Subject to the caveat that our model leaves out known nonlinear properties of MEC stellate cells (e.g., the slow inward rectifier), our results suggest that the precise channel numbers are relevant not only for determining neuronal reliability and threshold but also for the qualitative form of dynamics expressed by the neuron. For example, the maps of Fig. 2 show that the level of channel noise can determine the number of qualitative states that an excitable cell can enter; unlike its counterparts, the stochastic family of models with N = 4,800 can generate physiologically relevant Region 3 behavior robustly. This result implies that mismatches between the dynamics of deterministic models and actual neuronal response in some cases may result not from any inaccuracies in the model, but rather from a failure to incorporate the effects of stochastic channel gating. In stellate neurons of the MEC, noise induced by channel flicker is large enough to alter cellular inputoutput properties in response to both constant and timevarying stimuli. If this effect is indeed important, the statistical properties of voltagegated ion channels may exert an effect on the dynamics of neuronal networks.
Acknowledgments
We thank C. Chow, J. Collins, J. Rinzel, and A. Sherman for valuable discussions and F. Moss for reading an earlier version of this manuscript. Some of the currentclamp data were collected and shared graciously by B. Haman.
This work was supported by grants from the National Institutes of Health and the Whitaker Foundation to J. A. White, the Medical Research Council to A. Alonso, and the National Institute of Neurological Disorders and Stroke and the Office of Naval Research to A. R. Kay.
Footnotes

Address reprint requests to: J. A. White, Dept. of Biomedical Engineering, Boston University, 44 Cummington St., Boston, MA 02215.
 Copyright © 1998 the American Physiological Society