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Department of Physiology, University of Cambridge, Cambridge CB2 3EG, United Kingdom
Submitted 3 February 2004; accepted in final form 20 May 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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). The discharge frequencies of this latter type lay within a narrow range, clearly distinct from zero. From these observations, and previous work on other preparations, such as Cancer pagurus axons (Arvanitaki 1936) and decalcified nerves of frogs and squids (Brink et al. 1946), Hodgkin separated excitable membranes into two "classes," which have more recently been referred to as "types." The crucial distinction is that, whereas type 1 neurons show a continuous transition from zero frequency to arbitrarily low frequencies of firing, type 2 neurons show an abrupt onset of repetitive firing at a nonzero firing frequency. Clearly, all continuously firing neurons with a threshold will fall into one of these 2 types, which thus represent the behavior of a wide range of excitable membranes.
Even simple dynamical models of spike generation can exhibit both kinds of behavior, depending on their parameters (Morris and Lecar 1981; Rinzel and Ermentrout 1998). In these models, because of the different natures of dynamical bifurcation at threshold, type 1 behavior is associated with all-or-nothing spikes, whereas type 2 behavior is associated with graded spike amplitude and subthreshold oscillations. Recently, modeling studies have shown that the threshold type of the neuron profoundly affects the reliability of spike generation in the presence of noise (Gutkin and Ermentrout 1998; Robinson and Harsch 2002). Experimental classification of the responses of neurons in the cortex, however, has focused mostly on the form of the frequency vs. current (f–I) relationship in responses that are well above threshold (Connors and Gutnick 1990; Kawaguchi and Kubota 1997; Nowak et al. 2003); a clear classification of the continuity or discontinuity of the f–I relationship at threshold is lacking. Therefore in this paper we study the thresholds of 2 well-characterized types of cell—regular-spiking and fast-spiking neurons—and show that they follow type 1 and type 2 behaviors, respectively. We discuss what impact this could have on the roles of these 2 cell types in the cortical network.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Transverse slices were prepared from somatosensory cortex of 15- to 21-day-old Wister rats using standard techniques (Sakmann and Stuart 1995). During slicing, tissue was kept in sodium-free solution that had the following composition (in mM): 254 sucrose, 2.5 KCl, 26 NaHCO2, 10 glucose, 1.25 NaH2PO4, 2 CaCl2, and 1 MgCl2. Slices of 300 µm thickness were cut on a vibrating slicer (Microslicer DTK-3000, D.S.K., Kyoto, Japan) and kept in Ringer solution at room temperature for 
2 h before recording. The Ringer solution contained (in mM):125 NaCl, 2.5 KCl, 25 NaHCO2, 25 glucose, 1.25 NaH2PO4, 2 CaCl2, and 1 MgCl2. Both slicing and recording solutions were equilibrated with 95% O2,-5% CO2 gas to a final pH of 7.4. Slices were viewed with an upright microscope (Olympus BW50WI, Olympus UK, London) using infrared differential interference contrast optics. All experiments were performed at 30 ± 1°C. Whole cell patch-clamp recordings were made from the somas of neurons in layers 2/3. Putative regular-spiking cells were of pyramidal morphology, whereas putative fast-spiking cells were selected on the basis of a nonpyramidal shape and multipolar dendrites (Connors et al. 1982). During recording, the slices were perfused continuously with Ringer solution in which 10 µM bicuculline, 10 µM CNQX, and 10 µM AP5 (Tocris Cookson, Bristol, UK) were included to block most intrinsic synaptic conductances. However, some activity and synaptic input appeared to persist because there were typically still small fluctuations (1–2 mV) in the membrane voltage at the current-clamp mode. Somatic patch-pipette recordings were made with a Multiclamp 700A amplifier (Axon Instruments, Foster City, CA) in current-clamp mode, correcting for prenulled liquid junction potential. Whole cell recording pipettes (Clark GC150T-7.5) with 3.9–4.3 M
were filled with the standard intracellular solution: 105 mM K gluconate, 30 mM KCl, 10 mM HEPES, 10 mM phosphocreatine Na2, and 0.3 mM Na-GTP, balanced to pH 7.3 with NaOH. Series resistance compensation was used.
For conductance injection (dynamic-clamp) stimulation (Robinson and Kawai 1993; Sharp et al. 1993), an SM-1 conductance injection amplifier (Cambridge Conductance, Cambridge, UK) was used to inject step commands of ohmic excitatory conductance (reversal potential –10 mV) or mixtures of excitatory and inhibitory shunting conductance (reversing at the resting potential). Signals were filtered at 5 kHz and sampled with 12-bit resolution at 20 kHz. For stochastic stimulation, the sum of a constant current (I0) and a white Gaussian noise current [
(t)] was used; i.e., the stimulus current I(t) is described by I(t) = I0 + 
(t), where is the noise intensity. I0 was set to be just below the spike threshold of each cell so that no spike occurred in the absence of the noise term. This parameter regime is referred to as "excitable," whereas the parameter regime in which repetitive firing occurs is referred to as "oscillatory." In each run, a 10-s stimulus I(t) was repeatedly applied to cells 20 to 30 times, separated by 40-s recovery intervals, but each noise realization was different. In a test for temperature sensitivity, we measured from 6 cells (3 RS, 3 FS) at temperatures of 30, 34, and 37°C, and found no significant difference in critical frequency (fc) and maximal firing frequency (fmax) in any cell. Thus variations in temperature over this range did not seem to be a major factor shaping the threshold dynamics.
Spike statistics
Action potential shape parameters were measured from action potentials evoked by just-suprathreshold 200-ms current steps from a resting membrane potential near –70 mV. Spike amplitude was measured as the difference between the peak and the threshold of the action potential. Spike threshold was defined as the potential at which the first derivative of the voltage waveform exceeded 8 times its baseline SD. The afterhyperpolarization (AHP) was measured as the difference between the spike threshold and voltage minimum following the action potential peak. Spike width was measured at half the spike amplitude. Spike times were measured as the times of upward zero crossing of the membrane potential. Instantaneous frequency (reciprocal of interspike interval) was computed from trains of action potentials evoked by 600-ms duration pulses for the 1st, 2nd, and 4th interspike intervals. Steady-state (SS) firing frequency was computed as the average of instantaneous frequency for the last 5 intervals of a train. Current or conductance strength was usually progressively increased or decreased in small (10-pA or 500-pS) steps. Initial instantaneous frequency and SS firing rate were plotted as a function of the injected current or conductance strength, to construct frequency–current (f–I) or frequency–conductance (f–g) relationships. f–I relationships were fitted to the simple function f = (aI – b)d (I > b/a, 0 < d 
1), where f and I respectively represent firing frequency and current intensity and a, b, and d are constant parameters. A similar function was used by Ermentrout (1998). When d is close to 1, the f–I relationship is linear; if d is close to 0, it becomes markedly nonlinear (sublinear). The maximum firing rate of a neuron was computed from the number of spikes per trial at the highest current strength before depolarization block. The frequency adaptation properties of neurons were characterized by calculating the instantaneous firing rate as a function of time since the beginning of the 600-ms pulse. For each current intensity, the decay of firing rate was fitted to a single exponential function
 | (1) |
a, and Fa are positive constant parameters. Fa represents the adapted firing rate. The strength of adaptation (adaptation index, A) was quantified as 100 x (1 – Fa/F1), where F1 corresponds to the firing rate of the 1st interspike interval. For a given neuron, because the parameters of the firing rate curve strongly depend on the current intensity, the highest current level not producing depolarization block of spiking was used for comparison among cells. For some cells, no adequate exponential fit could be obtained. For these cells, Fa was calculated as the mean firing rate for the last 50 ms of the 600-ms current pulse and used to calculate the adaptation index. Results are reported as means ± SD. Estimating membrane impedance and frequency characteristics
Membrane impedance characteristics were estimated in both the time domain and the frequency domain. In the time domain, membrane time constants were obtained by fitting a single exponential function to the initial part of more than 10 time-averaged voltage responses to small (–20 or –10 pA), 600-ms-long hyperpolarizing current pulses. Input resistance was calculated from Ohm's law by dividing the maximal average voltage deflection by the amplitude of the applied current pulses. In the frequency method, a 20-s-long sinusoidal current with a frequency varying linearly from 0 to 32 or 128 Hz (ZAP function) was used as the stimulus (Gutfreund et al. 1995), and a maximal amplitude was set just below the threshold for producing spikes. To estimate the magnitude of the impedance, the current (I) and voltage (V) recordings were converted to the frequency domain by fast Fourier transform. Impedance (Z) was calculated from the coefficients of Fourier transforms. In complex polar notation, the impedance is expressed in terms of its magnitude |Z(f)| and phase 
(f)
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(f)] of the complex-valued impedance were plotted against frequency to give impedance-magnitude (IM) and impedance-phase (IP) profiles respectively. Subthreshold IM and IP profiles of individual neurons were compared with those expected for a passive membrane circuit with the same input resistance and time constant
 | (3) |
is the membrane time constant, and
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Input resistance and membrane time constant were calculated by fitting the impedance magnitude–frequency relationship using a standard method of least squares (function lsqcurvefit in Optimization Toolbox, MATLAB, The MathWorks, Natick, MA).
Numerical calculation of neural models
f–I curves of neural models were constructed with AUTO (Doedel and Kernevez 1986), a component of XPPAUT software (Ermentrout 2002). To simulate neural models in an excitable regime driven by white Gaussian noise, we used the forward improved Euler or Heun method (Kloeden 1999), with a time step of 
t = 1 µs. This method gives a higher-order discretization error than that of the simple Euler method. The stimulus current I(t) = I0 + (t) was set in the excitable regime, as described above for the experiments. To calculate coefficient of variation (SD/mean) of interspike intervals, a fixed-voltage threshold was used to detect spikes, but the exact level used had no significant effects on the results.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Based on responses to current-step injection, cells recorded in layer 2 or 3 of somatosensory cortex were classified into 2 groups: regular-spiking (RS) and fast-spiking (FS) cells. RS cells had a typical pyramidal morphology under infrared differential interference contrast optics (Fig. 1A), whereas FS cells were selected on the basis of a nonpyramidal shape and a round soma with multipolar dendrites (Fig. 2A). This study is based on recordings from 20 RS neurons and 23 FS neurons.
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and a time constant of 41.2 ± 12.5 ms estimated in the time domain (see METHODS). When estimated in the frequency domain (see METHODS) (Fig. 1C), the average input resistance was 349.7 ± 165.2 M and the average time constant was 40.2 ± 10.5 ms. RS neurons showed long-duration action potentials (1.71 ± 0.36 ms) and small AHPs (7.3 ± 2.4 mV), as shown in Fig. 1B. In response to sustained current injection, RS cells showed a relatively low maximum frequency firing (32.3 ± 7.0 Hz). In addition, RS cells showed substantial spike broadening between the 1st and 2nd action potentials of a train, with an average 2nd spike width of 2.22 ± 0.50 ms.
Figure 2B shows spike responses of FS neurons. These had an average resting potential of –71.2 ± 4.0 mV. They had an average input resistance of 357.4 ± 147.5 M and a time constant of 29.7 ± 5.9 ms when estimated in the time domain, and input resistance of 229.3 ± 73.1 M and time constant of 21.8 ± 9.2 ms estimated in the frequency domain. One of the reasons for the relatively larger discrepancies between time and frequency domain estimates of membrane parameters in FS cells is that the subthreshold impedance characteristics of FS cells could not always be well modeled by an RC circuit. They also had shorter-duration action potentials (1.18 ± 0.17 ms, P < 0.001) and larger AHPs (17.1 ± 4.1 mV, P < 0.001) than those of RS cells. FS cells had a much higher maximum frequency firing (61.0 ± 9.1 Hz, P < 0.001). They showed no substantial spike broadening between the 1st and 2nd action potentials of a train: the average 2nd spike width was 1.25 ± 0.18 ms, shorter than that of RS cells (P < 0.001).
A scatter-plot comparison of spike-shape parameters (AHP, the 2nd spike width, and A) versus maximal firing frequency for all cells revealed bimodal distributions of parameters, with little overlap between RS and FS cells, as shown in Fig. 3. Thus RS and FS neurons were reliably distinguished from each other on the basis of firing frequency, adaptation, and spike shape. The statistics for each cell type, together with significance levels for the differences between types, are shown in Table 1.
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In response to sustained current injection, RS cells showed a strong frequency adaptation (A = 71.4 ± 4.2) with a decay time constant (a) of 29.5 ± 9.5 ms, as shown in Fig. 4A. Figure 4B shows that their f–I relationship is continuous at threshold. The lowest frequencies approached the lowest measurable limit given the duration of the stimulus (1.66 Hz). In general, values slightly higher than this limit are expected, given the finite current increments used. For the 1st instantaneous and SS firing frequency versus current relationships, the average powers of the dependency of f on I [i.e., the exponent d in a fit to the relationship f = (aI – b)d; see METHODS] were respectively 0.84 ± 0.12 and 0.57 ± 0.12. The 1st interspike interval presumably reflects the instantaneous firing rate before substantial activation of an adaptation mechanism that has a nonlinear dependency on I. Thus the 1st instantaneous rate has a weaker nonlinearity than the SS firing f–I relationship.
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a) of 54.2 ± 33.8 ms (P < 0.05) (Fig. 4C). They were able to fire at high frequencies with relatively little spike frequency adaptation, which occurred mainly over the first few spikes. They also showed an abrupt onset of repetitive firing with increasing current, as shown in Fig. 4D. There is a clear discontinuity at a critical frequency fc in their f–I relationship at threshold. FS neurons are unable to support maintained regular firing below the critical frequency, which varied between 10 and 30 Hz for different cells. Above fc, the instantaneous firing rate increased monotonically with current strength. In some cases, however, the relation reached a clear plateau at current strengths below those that caused spike failure. For the 1st instantaneous and SS f–I relationships, average d values of fits were 0.65 ± 0.13 and 0.36 ± 0.12, both significantly smaller than for RS cells (P < 0.001). Note that the maximum firing frequency for FS cells (100 Hz) is lower than that reported in some other studies (Erisir et al. 1999; Kawaguchi 1995). We ascribe this to the comparatively young age of slices used here. Firing frequency and conductance input
During the normal operation of cortical neurons in the intact animal, the stimulus consists of the conductance of synaptic receptor channels, a current source that reacts to the membrane potential, and which can greatly alter the input resistance and membrane time constant of cells (Destexhe et al. 2001) and cause considerable shunting of action potential amplitude. To test whether this affects the classification of RS and FS cells' threshold behaviors, we used the conductance injection or dynamic-clamp technique (Robinson and Kawai 1993; Sharp et al. 1993) to stimulate action potential firing by step commands of ohmic excitatory conductance (reversal potential –10 mV) or mixtures of excitatory and inhibitory shunting conductance (reversing at the resting potential). Figure 5 shows responses of RS neurons to conductance stimulation. Either AMPA-type conductance alone (Fig. 5A, left) or in the presence of shunting, GABA-type conductance (Fig. 5A, right), produced an increasing firing frequency as the excitatory stimulus was increased. Interestingly, the firing frequency–conductance (f–g) relationship of instantaneous firing frequency showed a pronounced shift to the right with increasing shunting (Fig. 5B), whereas the SS f–g relationship was much less sensitive (Fig. 5C) at higher levels of excitation, possibly reflecting the activation of an additional intrinsic conductance, attenuating the effect of the injected GABA-type conductance. As with current stimulation, repetitive firing was supported at frequencies as low as could be assessed given the stimulus duration.
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Particularly with conductance stimulation, it was common in FS cells, but never in RS cells, to find intermittent switching between periodic firing at 20–30 Hz and silence on the edge of threshold (Fig. 6A, right, middle trace). This was also observed over a narrow range with current stimulation. Such irregular firing was excluded from f–g or f–I relationships, which are restricted to firing that is periodic throughout the test pulse period. This type of firing is a further indication of a type 2 discontinuity in the frequency–stimulus relationship at threshold.
The interruptions in periodic spiking during this unstable irregular bursting of FS neurons were clearly associated with small subthreshold oscillations (Fig. 7A). These oscillations showed a strong peak in power spectral density, which matched the threshold spike frequency. An example is shown in Fig. 7B, where the subthreshold frequency was about 20 Hz. In 13 different cells, the relationship between spiking and subthreshold oscillation was usually 1:1, although 2:1 oscillations were occasionally observed (Fig. 7C).
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It is widely appreciated that the massive synaptic bombardment received by neurons in vivo represents a strong source of noise. We investigated whether such noisy input has a different effect on the 2 types of cells at firing threshold. For the sake of simplicity, we used a simple white-noise stochastic current input to examine the threshold behaviors of 2 types of neurons rather than a synaptic waveform-based stochastic input (Chance et al. 2002; Harsch and Robinson 2000; Mitchell and Silver 2003).
Figure 8A shows the responses of a type 1 cell driven by 3 intensity levels of the noisy current input ( = 10, 50, and 100 pA) with a constant (subthreshold) current (I = 85 pA) in the excitable regime, such that the cell is silent in the absence of the noise term. As the noise intensity increases, the average firing frequency monotonically increases from a very low firing frequency in all 5 cells, as shown in Fig. 8B. This result resembled the f–I curves shown in Fig. 4B. We quantified the degree of response variability of the type 1 neurons by measuring the coefficient of variation (CV) of the interspike intervals. Figure 8C shows average CV versus noise intensity plots for 5 different cells in their excitable regimes. In each case, CV first decreases with increasing , reaching a minimum at an optimum value, and then increases. Because smaller values of CV mean coherence of oscillations (note that in the case of periodic oscillations, CV = 0), such a minimum represents a "coherence resonance" (Pikovsky and Kurths 1997).
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= 10, 50, and 120 pA) with a constant current (I = 70 pA). At the lowest noise level in Fig. 9A, subthreshold oscillations predominate over suprathreshold spike activity (top panel). Although monotonically increasing, f– relationships for 5 cells were discontinuous at low frequencies and flatter than those of RS cells (Fig. 9B). Such properties are also reminiscent of the f–I curves shown in Fig. 4D, but blurred by the stochastic input. Figure 9C shows average CV versus noise intensity plots for 5 different cells in their excitable regimes. In each case, there is a decrease in CV toward a minimum with increasing , followed by a slight increase at high . Thus coherence resonance also occurs in the type 2 case, but oscillations were more coherent over a wider range of noise intensities.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Nowak et al. 2003), the emphasis has been on mapping the entire relationship, and the current increments used around threshold have not exposed the qualitative distinction between RS and FS neurons. However, Kawaguchi and Erisir et al. reported an "abrupt onset" of firing in FS neurons (Erisir et al. 1999; Kawaguchi 1995). In the present study, the granularity of the stimulus current steps was set as fine as possible, consistent with obtaining repeatable results from trial to trial. A further constraint was the recording time required for very fine increments in stimulus. We also showed that the same type of relationship between stimulus and firing frequency holds for conductance input, where the natural effects of shunting and reduced time constant associated with synaptic input are reproduced, over a range of levels of background inhibition. Background inhibitory conductance caused a rightward shift in the f–g relationship, as would be expected for static conductance input (Mitchell and Silver 2003). In addition, we have shown that the same type of relationship holds for white Gaussian noise current input, where the basic features of the noise-free f–I relationship are retained, but blurred, in the f– relationship.
A number of theoretical studies have characterized bifurcations of neural models from the standpoint of the geometrical theory of (nonlinear) dynamical systems (Izhikevich 2000 and references therein). Bifurcation describes a qualitative change in dynamics that can be observed as a system parameter varies (Guckenheimer and Holmes 1983). The difference between type 1 and type 2 is well understood in dynamical models of spike generation (Fig. 10). For example, the Connor et al. model of molluscan neurons shows a type 1 f–I relationship (Fig. 10A), a consequence of the A-type K+ current that activates rapidly near to rest (Connor et al. 1977), whereas the Hodgkin–Huxley model of squid giant axon (Hodgkin and Huxley 1952) has type 2 behavior (Fig. 10B). These models have 6 and 4 dynamical variables, respectively, but the essence of type 1 and type 2 behavior may be understood with 2 variables, as in the FitzHugh–Nagumo (FitzHugh 1961; Nagumo et al. 1962) or Morris–Lecar (Morris and Lecar 1981) models, in which one variable represents voltage and Na+ or Ca2+ channel activation, and one represents Na+ channel inactivation/K+ channel activation. At a particular stimulus level, each of these variables has a voltage-dependent derivative, defining a 2-D vector field in the phase plane.
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and by Fellous et al. (2001).
Thus this is a further reason for our classification of FS neurons as type 2. Although the bifurcations described above are the most typical ones encountered in lower-dimensional neural models, the mechanism of transition from rest to oscillatory activity in neurons is not restricted to these. Our aim here is not to identify the precise type but to illustrate the feasibility of such bifurcations in lower-dimensional models. Because higher-dimensional spiking models may often be accurately reduced into fast and slow systems of variables, such type 1 and 2 threshold bifurcations may apply quite generally to biological neurons. The noise sensitivity and variability of interspike intervals also strongly depend on the dynamical type. Figure 11 shows CVs of interspike intervals of the ML model driven by white Gaussian noise input on the I0– plane, for type 1 (Fig. 11A) and type 2 (Fig. 11B) parameters. In the excitable regime (I < 40 µA/cm2) the type 1 ML model shows relatively large CVs (>0.4), whereas it exhibits coherent oscillations (CV < 0.2) only in the oscillatory regime (I > 40 µA/cm2) with small noise (0 < < 2 µA/cm2). In contrast, the type 2 ML model shows smaller CVs over a larger region of the I0– plane (Fig. 11B), demonstrating that coherent oscillations are easily achieved over a wide range of noise intensities even in the excitable regime (I < 88 µA/cm2).
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; Schneidman et al. 1998)—the variability of neuronal firing will be dominated by the behavior at threshold. Type 1 and 2 models have very different noise sensitivities (Gutkin and Ermentrout 1998; Robinson and Harsch 2002). Fluctuation of the stimulus level below threshold type 1 neurons—as their dynamics slow down critically at the threshold—easily and rapidly allows spike generation to become effectively noise-dominated (Poisson).
What is the likely relevance of this to the operation of the cortex? Pyramidal RS cells appear to associate inputs from different layers and areas in the cortex, by the back-propagation–activated dendritic calcium-spike mechanism (Larkum et al. 1999). Type 1 behavior of RS neurons might allow relatively easy switching between different tempos in their inputs. It may also promote the generally high level of firing variability in the cortex, as suggested by Gutkin and Ermentrout (1998)—probably over 50% of cortical cells are type 1 RS neurons. RS neurons are, by virtue of their type 1 dynamics, suited to encoding even low stimulus levels in the firing rate (i.e., as randomized, Poisson-process–like firing); this may have advantages for network stability, by breaking up exact synchrony.
On the other hand, type 2 FS neurons, which inhibit each other and other RS neurons locally (Holmgren et al. 2003), are implicated in promoting episodes of synchronous firing (Beierlein et al. 2000; Galarreta and Hestrin 2001). They are coupled together by electrical synapses or gap junctions, which helps to synchronize their action potentials precisely (Gibson et al. 1999). The nature of type 2 dynamics may mean that the phase of rhythmic firing is quite stable even when the mean stimulus goes subthreshold because strong subthreshold oscillations could keep the rhythm intact until the stimulus moves above threshold again: there is fast motion below threshold. For a type 2 neuron, there is a much greater tendency for spikes to drop out without a consequent wide dispersion of spike times as the stimulus passes below threshold (Robinson 2004). In other words, type 2 neurons intrinsically prefer to stay coherent with their input or to be silent, whereas type 1 neurons have a graded transition between the 2 extremes. The identity of subthreshold oscillation and suprathreshold spike frequency (fc) for FS neurons could lead to stable synchronous oscillations at 20–30 Hz, as have been suggested to be important in sensory feature recognition and binding (Singer and Gray 1995). The increase in fc produced by GABAergic input, as shown in Fig. 6C, could provide a mechanism for a physiological modulation of this synchronous firing frequency.
Computational modeling provides some support for the scenarios described above; the dynamics of coupled neurons and of neural networks critically depend on their excitability types. For example, Hansel et al. (1995) showed that excitatory synapses cannot lead to synchronization for type 1 neural models like the Connor et al. model unless the synapses are very fast. Hansel et al. also showed numerically that type 2 excitability can easily lead to synchronization of coupled oscillations, using the Hodgkin–Huxley model. In addition, Ermentrout (1996) reported that, compared with type 2 models, synchrony is in general difficult to achieve when oscillators have type 1 excitability.
In conclusion, there are two types of threshold behavior for periodically firing neurons, which show continuous (type 1) or discontinuous (type 2) f–I curves. We have shown that in the cortex, regular-spiking and fast-spiking neuronal types have type 1 and type 2 thresholds, respectively. This endows them with fundamentally different spike rate encoding, noise sensitivity, subthreshold, and synchronization properties. Thus they are expected to serve very different roles in the dynamics of the cortical network.
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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Present affilitation and address for reprint request: T. Tateno, Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.
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REFERENCES |
|---|
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Beierlein M, Gibson JR, and Connors BW. A network of electrically coupled interneurons drives synchronized inhibition in neocortex. Nat Neurosci 3: 904–910, 2000.[CrossRef][ISI][Medline]
Brink F, Bronk DW, and Larrabee MG. Chemical excitation of nerve. Ann NY Acad Sci 47: 457–485, 1946.[CrossRef][ISI]
Connor JA, Walter D, and McKown R. Modifications of the Hodgkin–Huxley axon suggested by experimental results from crustacean axons. Biophys J 18: 81–102, 1977.
Connors BW and Gutnick MJ. Intrinsic firing patterns of diverse neocortical neurons. Trends Neurosci 13: 99–104, 1990.[CrossRef][ISI][Medline]
Connors BW, Gutnick MJ, and Prince DA. Electrophysiological properties of neocortical neurons in vitro. J Neurophysiol 48: 1302–1320, 1982.
Destexhe A, Rudolph M, Fellous J-M, and Sejnowski TJ. Fluctuating synaptic conductances recreate in-vivo-like activity in neocortical neurons. Neuroscience 107: 13–24, 2001.[CrossRef][ISI][Medline]
Doedel E and Kernevez JP. AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations, Applied Mathematics Report. Pasadena, CA: California Institute of Technology, 1986.
Erisir A, Lau D, Rudy B, and Leonard CS. Function of specific K channels in sustained high-frequency firing of fast-spiking neocortical interneurons. J Neurophysiol 82: 2476–2489, 1999.
Ermentrout B. Type 1 membrane, phase resetting curves, and synchrony. Neural Comput 8: 979–1002, 1996.[Abstract]
Ermentrout B. Linearization of F–I curves by adaptation. Neural Comput 10: 1721–1729, 1998.[Abstract]
Ermentrout B. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2002.
Fellous JM, Houweling AR, Modi RH, Rao RPN, Tiesinga PHE, and Sejnowski TJ. Frequency dependence of spike timing reliability in cortical pyramidal cells and interneurons. J Neurophysiol 85: 1782–1787, 2001.
FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1: 445–466, 1961.
Galarreta M and Hestrin S. Spike transmission and synchrony detection in networks of GABAergic interneurons. Science 292: 2295–2299, 2001.
Gibson JR, Beierlin M, and Connors BW. Two networks of electrically coupled inhibitory neurons in neocortex. Nature 402: 75–79, 1999.[CrossRef][Medline]
Guckenheimer J and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Berlin: Springer-Verlag, 1983.
Gutfreund Y, Yarom Y, and Segev I. Subthreshold oscillations and resonant frequency in guinea-pig cortical neurons: physiology and modelling. J Physiol 483: 621–640, 1995.[ISI][Medline]
Gutkin B and Ermentrout B. Dynamics of membrane excitability determine interspike interval variability: a link between spike generation and cortical spike train statistics. Neural Comput 10: 1047–1065, 1998.[Abstract]
Hansel D, Mato G, and Meunier C. Synchrony in excitatory neural networks. Neural Comput 7: 307–337, 1995.[Abstract]
Hodgkin AL. The local electric changes associated with repetitive action in a non-medullated axon. J Physiol 107: 165–181, 1948.
Hodgkin AL and Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 463: 391–407, 1952.
Holmgren C, Harkany T, Svennenfors B, and Zilberter Y. Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. J Physiol 551: 139–153, 2003.
Izhikevich EM. Neural excitability, spiking and bursting. Int J Bifurcation Chaos 10: 1171–1266, 2000.[CrossRef][ISI]
Kawaguchi Y. Physiological subgroups of nonpyramidal cells with specific morphological characteristics in layer II/III of rat frontal cortex. J Neurosci 15: 2638–2655, 1995.[Abstract]
Kawaguchi Y and Kubota Y. GABAergic cell subtypes and their synaptic connections in rat frontal cortex. Cereb Cortex 7: 476–486, 1997.
Kloeden PE and Platen E. Numerical Solution of Stochastic Differential Equations. Berlin: Springer-Verlag, 1999.
Larkum ME, Kaiser KMM, and Sakmann B. Calcium electrogenesis in distal apical dendrites of layer 5 pyramidal cells at a critical frequency of back-propagating action potentials. Proc Natl Acad Sci USA 96: 14600–14604, 1999.
Lecar H and Nossal R. Theory of threshold fluctuations in nerves. I. Relationships between electrical noise and fluctuations in axon firing. Biophys J 11: 1048–1067, 1971.
Llinas RR, Grace AA, and Yarom Y. In vitro neurons in mammalian cortical layer 4 exhibit intrinsic oscillatory Activity in the 10- to 50-Hz frequency range. Proc Natl Acad Sci USA 88: 897–901, 1991.
Mitchell SJ and Silver RA. Shunting inhibition modulates neuronal gain during synaptic excitation. Neuron 38: 433–445, 2003.[CrossRef][ISI][Medline]
Morris C and Lecar H. Voltage oscillations in the barnacle giant muscle fibre. Biophys J 35: 193–213, 1981.
Nagumo JS, Arimoto S, and Yoshizawa S. An active pulse transmission line stimulating a nerve axon. Proceedings IRE 50: 2061–2070, 1962.
Nowak LG, Azouz R, Sanchez-Vives MV, Gray CM, and McCormick DA. Electrophysiological classes of cat primary visual cortical neurons in vivo as revealed by quantitative analyses. J Neurophysiol 89: 1541–1566, 2003.
Pike FG, Goddard RS, Suckling JM, Ganter P, Kasthuri N, and Paulsen O. Distinct frequency preferences of different types of rat hippocampal neurones in response to oscillatory input currents. J Physiol 529: 205–213, 2000.
Pikovsky AS and Kurths J. Coherence resonance in a noise-driven excitable system. Phys Rev Lett 78: 775–778, 1997.[CrossRef]
Rinzel J and Ermentrout B. Analysis of neural excitability and oscillations. In: Methods in Neuronal Modelling: From Ions to Networks, edited by Koch C and Segev I. Cambridge, MA: MIT Press, 1998, p. 251–291.
Robinson HPC. The biophysical basis of firing variability in cortical neurons. In: Computational Neuroscience: A Comprehensive Approach, edited by Feng JF. London: CRC LLC, 2004, p. 159–183.
Robinson HPC and Harsch A. Stages of spike time variability during neuronal responses to transient inputs. Phys Rev E 66: 061902, 2002.[CrossRef]
Robinson HPC and Kawai N. Injection of digitally synthesized synaptic conductance transients to measure the integrative properties of neurons. J Neurosci Methods 49: 157–165, 1993.[CrossRef][ISI][Medline]
Sakmann B and Stuart G. Patch-pipette recordings from the soma, dendrites and axon of neurons in brain slices. In: Single-Channel Recording, edited by Sakmann B and Neher E. New York: Plenum, 1995, p. 199–211.
Schneidman E, Freedman B, and Segev I. Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comput 10: 1679–1703, 1998.[Abstract]
Sharp AA, O'Neil MB, Abbott LF, and Marder E. Dynamic clamp: computer-generated conductances in real neurons. J Neurophysiol 69: 992–995, 1993.
Singer W and Gray CM. Visual feature integration and the temporal correlation hypothesis. Ann Rev Neurosci 18: 555–586, 1995.[CrossRef][ISI][Medline]
Tateno T and Pakdaman K. Random dynamics of the Morris–Lecar neural model. Chaos: An Interdisciplinary Journal of Nonlinear Science 14: 511–530, 2004.[CrossRef]
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20 spikes/s). SS frequency reaches a plateau at about 350 pA.
