During wakefulness, pyramidal neurons in the intact brain are bombarded by synaptic input that causes tonic depolarization, increased membrane conductance (i.e., shunting), and noisy fluctuations in membrane potential; by comparison, pyramidal neurons in acute slices typically experience little background input. Such differences in operating conditions can compromise extrapolation of in vitro data to explain neuronal operation in vivo. For instance, pyramidal neurons have been identified as integrators (i.e., class 1 neurons according to Hodgkin's classification of intrinsic excitability) based on in vitro experiments but that classification is inconsistent with the ability of hippocampal pyramidal neurons to oscillate/resonate at theta frequency since intrinsic oscillatory behavior is limited to class 2 neurons. Using long depolarizing stimuli and dynamic clamp to reproduce in vivo-like conditions in slice experiments, we show that CA1 hippocampal pyramidal cells switch from integrators to resonators, i.e., from class 1 to class 2 excitability. The switch is explained by increased outward current contributed by the M-type potassium current IM, which shifts the balance of inward and outward currents active at perithreshold potentials and thereby converts the spike-initiating mechanism as predicted by dynamical analysis of our computational model. Perithreshold activation of IM is enhanced by the depolarizing shift in spike threshold caused by shunting and/or sodium channel inactivation secondary to tonic depolarization. Our conclusions were validated by multiple comparisons between simulation and experimental data. Thus even so-called “intrinsic” properties may differ qualitatively between in vitro and in vivo conditions.
The high level of background synaptic input received by pyramidal neurons in the intact, awake brain can significantly alter the integrative properties of those neurons (Bernander et al. 1991; Destexhe and Paré 1999; Paré et al. 1998). Destexhe and Paré (1999) reported that, on average, background synaptic input to neocortical pyramidal neurons reduced input resistance by 80% while causing a 20 mV depolarizing shift in membrane potential and a tenfold increase in the SD of voltage noise (for review, see Destexhe et al. 2003). Reducing input resistance (i.e., shunting) causes predictable shortening of both the membrane time and length constants; this in turn causes the cell to behave more like a coincidence detector and less like an integrator insofar as the neuron responds more selectively to synchronized inputs (Bernander et al. 1991; Rudolph and Destexhe 2003). Thus passive membrane properties differ between in vitro and in vivo conditions with important consequences for how neurons process information. Noise caused by synaptic input has equally unintuitive implications, for instance, on firing rate modulation by shunting inhibition (Chance et al. 2002; Prescott and De Koninck 2003) and on contrast invariance of orientation tuning in visual cortex (Anderson et al. 2000). Wolfart et al. (2005) showed in thalamic neurons that conductance noise blurs the single-spike and burst firing modes (which are classically associated with depolarized and hyperpolarized membrane potentials, respectively) and interacts with intrinsic currents to determine the neuron's transfer function. Steriade (2001) described several examples in which noise influences “intrinsic” spiking patterns. Still other discrepancies are likely to exist between in vitro and in vivo conditions.
Here we address whether CA1 pyramidal neurons exhibit class 1 or class 2 excitability according to Hodgkin's classification scheme (Hodgkin 1948): class 1 neurons can maintain arbitrarily slow firing and therefore have a continuous frequency- current (f-I) curve, whereas class 2 neurons cannot maintain firing below some critical rate and therefore have a discontinuous f-I curve. Subsequent work has shown that this phenomenological difference derives from the different dynamical mechanisms responsible for spike initiation in each cell class (Izhikevich 2007; Prescott et al. 2008; Rinzel and Ermentrout 1998). The spike-initiating mechanism, in turn, has several important implications. For instance, the phase response curve (PRC) differs between class 1 and 2 neurons (Rinzel and Ermentrout 1998): a perturbation can only advance the next spike in class 1 neurons (i.e., monophasic PRC), whereas it can advance or delay the next spike in class 2 neurons depending on when the perturbation occurs (i.e., biphasic PRC). This property affects the synchronization of neurons within a network: class 1 neurons will not synchronize in a purely excitatory network, whereas class 2 neurons will (Ermentrout 1996; Hansel et al. 1995), although class 1 neurons can be synchronized by inhibitory input (e.g., Wang and Buzsáki 1996). Moreover, because of their spike-initiating mechanism, class 1 neurons summate inputs across a broad range of frequencies, whereas class 2 neurons have a preferred input frequency to which they resonate. Class 1 and 2 neurons have therefore been labeled integrators and resonators, respectively (Izhikevich 2007).
It seems to be commonly believed that pyramidal cells are class 1, or at least that is how they have been modeled (e.g., Wilson 1999) and the assumption is implicit in the widespread use of leaky integrate-and-fire models. But there is an unrecognized discrepancy: hippocampal CA1 pyramidal neurons do indeed have continuous f-I curves according to in vitro experiments (e.g., Gustafsson and Wigstrom 1981), consistent with class 1 excitability, but they can (under slightly different conditions) also exhibit subthreshold oscillations and resonance at 3–10 Hz or θ frequency (Hu et al. 2002; Leung and Yu 1998; Leung and Yim 1991; Pike et al. 2000), consistent with class 2 excitability. Needless to say, network oscillations are important for brain function (Buzsáki 2006) and although intrinsic resonance is not required for entrainment of pyramidal cell activity, it influences responsiveness to oscillatory input and is likely to affect θ-related phase precession (Harris et al. 2002). It is therefore critical that we understand and resolve this discrepancy so that neurons are modeled with the appropriate spike-initiating mechanism.
We reasoned that synaptic bombardment experienced in vivo causes a relative increase in outward current via shunting and activation or inactivation of currents secondary to tonic depolarization. Based on our recent work on spiking initiating dynamics (Prescott et al. 2008), a relative increase in outward current could theoretically convert neuronal excitability from class 1 to class 2. In other words, CA1 pyramidal cells that behave as integrators under in vitro conditions may behave as resonators under in vivo conditions. To investigate this, we first tested the effects of shunting and adaptation in a computational model amenable to dynamical analysis. Then, directed by specific predictions derived from modeling, we tested CA1 pyramidal neurons under different in vivo-like conditions. As we predicted, results demonstrate that “intrinsic” spike-initiating dynamics do indeed differ between in vitro and in vivo conditions because of the relative increase in outward current in vivo.
Slicing and electrophysiology
All experiments were done in accordance with regulations of the Canadian Council on Animal Care and have been described by us previously (Prescott et al. 2006). Adult male Sprague Dawley rats were anesthetized with intraperitoneal injection of sodium pentobarbital (30 mg/kg) and perfused intracardially with ice-cold oxygenated (95% O2-5% CO2) sucrose-substituted artificial cerebrospinal fluid (ACSF) containing (in mM) 252 sucrose, 2.5 KCl, 2 CaCl2, 2 MgCl2, 10 glucose, 26 NaHCO3, 1.25 NaH2PO4, and 5 kynurenic acid. The brain was rapidly removed and sectioned coronally to give 400-μm-thick slices, which were kept in normal oxygenated ACSF (126 mM NaCl instead of sucrose and without kynurenic acid) at room temperature until recording.
Slices were transferred to a recording chamber constantly perfused with oxygenated (95% O2-5% CO2) ACSF heated to 30–31°C. Pyramidal neurons in the CA1 region of hippocampus were recorded in the whole cell configuration with >80% series resistance compensation using an Axopatch 200B amplifier (Molecular Devices; Palo Alto, CA). Only data from regular spiking pyramidal cells judged healthy on the basis of three criteria (resting membrane potential less than −50 mV, spikes overshooting 0 mV, and input resistance >100 MΩ) were included in analysis presented here. Average resting V was −73 ± 1 (SE) mV, and average input resistance was 159 ± 9 MΩ. Variations in resting V were eliminated by adjusting V to −70 mV through tonic current injection. Reported values of V were corrected for liquid junction potential calculated (see Barry and Lynch 1991) to be 9 mV based on the intracellular recording solution, which contained (in mM) 135 KMeSO4, 5 KCl, 10 HEPES, and 2 MgCl2, 4 ATP (Sigma), 0.4 GTP (Sigma) as well as 0.1% Lucifer yellow; pH was adjusted to 7.2 with KOH. Pyramidal morphology was confirmed with epifluorescence after recording. All experiments were performed in 10 μM bicuculline methiodide (Research Biochemicals, Natick, MA), 10 μM 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX), and 40 μM d-2-amino-5-phosphonovaleric acid (d-AP-5, Tocris Cookson, Bristol, UK) to block residual background synaptic activity.
To recreate tonic depolarization, long depolarizing current steps were injected into the cell via the recording pipette; current magnitude is reported as Istim. To recreate increased membrane conductance (i.e., shunting), an artificial conductance was applied via dynamic clamp implemented with a Digidata 1200A ADC/DAC board (Molecular Devices) and DYNCLAMP2 software (Pinto et al. 2001) running on a dedicated processor; update rate was 10 kHz. For this study, we only report the effects of a single conductance level that was constant at 10 nS and was associated with a reversal potential of −70 mV; this conductance caused an average 56% reduction in input resistance. To recreate the voltage noise caused by synaptic input, noisy current (see following text) was injected into the cell, but the effects of noise were not analyzed in this study except for one experiment identified in results. Recreating the effects of background synaptic activity using dynamic clamp has been previously validated and discussed in detail elsewhere (Chance et al. 2002; Destexhe et al. 2001; Prescott et al. 2006; for review, see Prescott and De Koninck 2008).
Traces were low-passed filtered at 4 kHz and stored on videotape using a digital data recorder (VR-10B, Instrutech; Port Washington, NY). Off-line, recordings were sampled at 10 kHz on a computer using Strathclyde Electrophysiology software (J. Dempster, Department of Physiology and Pharmacology, University of Strathclyde, Glasgow, UK) and analyzed using locally designed software (De Koninck).
Simulations were based on a modified Morris-Lecar model (Morris and Lecar 1981; Rinzel and Ermentrout 1998) that we have analyzed in detail (Prescott et al. 2008). The model is described by the following equations (1) (2) (3) (4) (5) (6)
This minimal model was used instead of more complex, higher dimensional models (e.g., Hodgkin-Huxley) because it is amenable to dynamical analysis that can be used to explain key features of the model's behavior (see following text). For example, the Morris-Lecar model neglects sodium channel inactivation since it operates on the same time scale as delayed rectifier channel activation (e.g., Kepler et al. 1992). Our model is modified from the original Morris-Lecar model primarily by the addition of adaptation. All parameters were set according past modeling studies (Prescott et al. 2006, 2008) rather than being quantitatively fit to experimental data, notwithstanding the final set of simulations presented in Figs. 8 and 9.
Unless otherwise stated, ENa = 50 mV, EK = −100 mV, Eshunt = −70 mV, ḡNa = 20 mS/cm2, ḡK,dr = 20 mS/cm2, φw = 0.25, C = 2 μF/cm2, βm = −1.2 mV, γm = 18 mV, βw = −9 mV, and γw = 10 mV. Low- and high-conductance states were simulated by changing gshunt from 2 to 4 mS/cm2. In additional simulations not reported here, the 2 mS/cm2 increase in gshunt was associated with a reversal potential of −57 mV, which more accurately reflects the net reversal potential for background synaptic activity (see following text); results were qualitatively unchanged from simulations in which the reversal potential was −70 mV. Adaptation comprised two separate currents: M-type K+ current with ḡM = 2 mS/cm2, τz = 200 ms, βz = −30 mV, and γz = 5 mV and calcium-activated K+ current with ḡAHP = 1 mS/cm2, τz = 200 ms, βz = 0 mV, and γz = 5 mV. Activation of IM and IAHP were controlled separately by zM and zAHP.
Weak noise was included in all simulations investigating membrane potential oscillations (MPOs) to replicate noise that is still present in real neurons after blockade of fast synaptic input. This noise prevents the model from being unphysiologically stable by perturbing the system from its fixed point; MPOs are generated as the system returns toward its fixed point, meaning at least some noise is necessary for the generation of irregular MPOs like those observed experimentally (Dorval and White 2005; Erchova et al. 2004; Hutcheon and Yarom 2000). Noisy current was modeled as an Ornstein-Uhlenbeck process dInoise/dt = –Inoise/τnoise + σnoise N(0,1) where N is a random number with 0 mean and unit variance, and σnoise = 0.1 or 0.01 μA/cm2 when τnoise = 5 or 500 ms, respectively, to give equivalent variance in Inoise which equals σ2τ/2 (see Gillespie 1996). These values of σnoise are small compared with those used to replicate synaptic noise.
For the model, all conductances and currents are reported in densities (i.e., per unit area). For comparison with experiments in which, for example, Istim is not expressed per unit area, the model neuron can be assumed to have a surface area of about 300 μm2 (= 3 × 10−6 cm2). In other words, Istim in μA/cm2 multiplied by 3 × 10−6 cm2 × 106 pA/μA gives Istim in pA.
Our modified Morris-Lecar model is ideally suited for dynamical analysis techniques because it comprises the minimum number of variables (dimensions) required to reproduce the phenomena of interest. Ideally, the system can be simplified to two dimensions, i.e., the interaction between a fast activation variable (e.g., V) and a slower recovery variable (e.g., w or z). That interaction can be visualized by plotting the fast variable against the slower variable to create a phase portrait. Nullclines represent areas in phase space where a given variable remains constant. How the nullclines intersect (i.e., whether the intersection is stable or unstable) determines whether the system evolves toward a fixed point or toward a limit cycle (i.e., subthreshold membrane potential or repetitive spiking, respectively). Stability of the fixed point can be determined by local stability analysis, which involves linearizing the nullclines in the vicinity of their intersection and finding the eigenvalues of the Jacobian matrix. This analysis leads to inequalities discussed in Fig. 5 (see Borisyuk and Rinzel 2005 for full derivations) and helps explain how voltage trajectories evolve toward or away from the fixed point, including the frequency of subthreshold MPOs.
The original Morris-Lecar model is two dimensional (2D). Our model is 3D because of the addition of slow adaptation. Because adaptation develops slowly relative to the other variables, z (which controls adaptation; see preceding text) can be approximated as constant over some time window, thus allowing the model to be reduced from 3D to 2D; in other words, z is treated as a parameter and behavior is explained based on dynamical interaction between V and w. Initial analysis is based on this assumption. In the final two figures, however, slow changes in z are shown to be important for generating low-frequency oscillations. In this case, the system can still be approximated as 2D but with oscillations being explained by dynamical interaction between V and z (see results).
To determine the effects of certain parameters, the parameter of interest (e.g., Istim) can be continuously varied to determine its effects on the system's behavior, which will be reflected in the nullcline intersection. Of particular interest are bifurcations or abrupt transitions in the system's behavior, such as the transition between quiescence and repetitive spiking. See Rinzel and Ermentrout (1998) for a more detailed discussion of these methods in the context of neuronal excitability or Strogatz (1998) for an in-depth explanation.
Effects of shunting and adaptation on excitability in a modified Morris-Lecar model
Class 1 and 2 neurons initiate spikes through distinct dynamical mechanisms. Those dynamics represent competition between inward (depolarizing) and outward (hyperpolarizing) currents active at perithreshold potentials. In class 1 neurons, net current is inward at steady state. Sustained weak inward current allows the neuron to maintain slow repetitive spiking. This is manifested dynamically as a saddle-node on invariant circle (SNIC) bifurcation. In class 2 neurons, on the other hand, net current is outward at steady state. In this case, spike initiation requires that fast-activating inward current produces suprathreshold depolarization before slower-activating outward current becomes strongly activated (i.e., before net current becomes outward). This is manifested dynamically as a Hopf bifurcation. Importantly, class 2 neurons cannot maintain spiking below a critical rate lest slow-activating outward current overtake fast-activating inward current. For more detailed discussion of these points, see Izhikevich (2007) or Prescott et al. (2008).
A relative increase in outward current therefore encourages spike initiation through a Hopf bifurcation (class 2 excitability), whereas a relative increase in inward current encourages spike initiation through a SNIC bifurcation (class 1 excitability). Adaptation is caused by potassium currents the reversal potential of which is around −90 mV and, in the case of the M-type current IM, clearly contributes an outward current at perithreshold potentials. Similarly, shunting will produce an outward current at perithreshold potentials if its reversal potential is more hyperpolarized than threshold. Chance et al. (2002) calculated the conductance-weighted average of excitatory and inhibitory reversal potentials to be −57 mV, which is consistent with average membrane potentials reported by Destexhe and Paré (1999). Therefore shunting and adaptation both contribute outward current and are therefore predicted to encourage class 2 excitability. Decreased inward current (through cumulative sodium channel inactivation, for example) could cause a similar shift in net current with equivalent consequences for excitability (see following text).
To begin, we tested our theoretical prediction that increased outward current contributed by shunting and/or adaptation encourages class 2 excitability. First, we performed experiments in which in vivo-like conditions were replicated in acute brain slices using prolonged depolarizing current injections to activate adaptation with or without dynamic clamp to artificially increase membrane conductance (i.e., cause shunting) (Fig. 1 A, top). Then, starting with a modified Morris-Lecar model the parameters of which were set to values that give class 1 excitability, we added shunting and adaptation to reproduce the experimentally observed prohibition of repetitive spiking in the high-conductance state (Fig. 1A, bottom); the basis for this phenomenon has already been described in detail (Prescott et al. 2006) and results from nonlinearly increased activation of IM at perithreshold potentials in the high-conductance state.
With model parameters thus established, we used bifurcation analysis to ascertain the spike-initiating mechanism under different conditions. Under control conditions (i.e., without shunting or adaptation), spikes were generated through a SNIC bifurcation (Fig. 1Ba), which is indicative of class 1 excitability. Consistent with the predicted effects of an increase in outward current (see preceding text), the spike-initiating mechanism switched to a subcritical Hopf bifurcation when shunting and/or adaptation were included in the model (Fig. 1B, b–d), thus demonstrating conversion to class 2 excitability. The switch in bifurcation mechanism was paralleled by a change in the steady-state I-V curve, from nonmonotonic under control conditions (Fig. 1Ca) to monotonic when shunting and/or adaptation were added (Fig. 1C, b–d). A nonmonotonic curve means there is a local maximum above which depolarization activates net inward current at steady state—this is the biophysical basis for spike initiation through a SNIC bifurcation. A monotonic curve has no local maximum, meaning fast-activating inward current must cause suprathreshold depolarization before slower-activating outward current causes net steady-state current to become outward—this is the biophysical basis for spike initiation through a Hopf bifurcation (see preceding text). The change in shape of the steady-state I-V curve is therefore consistent with the relative increase in outward current contributed by shunting and/or adaptation which causes net current at perithreshold potentials to change direction (see following text).
Effects of increased outward current on voltage threshold
Shunting and adaptation both caused a depolarizing shift in spike threshold (V*) according to bifurcation analysis in Fig. 1B; threshold can be ascertained from voltage at the bifurcation. The shift is explained by the fact that inward current mediated by the fast sodium current INa must offset increased outward current contributed by shunting and adaptation; increased depolarization is required to activate INa more strongly to provide this increased inward current (Fig. 2 A) and contributes to the depolarizing shift in V* (Prescott et al. 2006). Importantly, the shift in V* occurs in a steep region of the voltage-dependent activation curve for the delayed rectifier potassium current IK,dr (Fig. 2B) and for IM (see following text). Consequently, a small change in V* allows dramatically increased activation of potassium currents at perithreshold potentials: this compounds the increase in outward current, exacerbates the depolarizing shift in V*, and causes the steady-state I-V curve to become monotonic (Fig. 2C).
Simulations therefore predicted that shunting and adaptation cause an increase in V*. We therefore measured V* in experimental data based on the voltage at which d2V/dt2 exceeded a cutoff value indicative of a spike. As predicted, both shunting and adaptation caused a depolarizing shift in V* (Fig. 3, A and B). Furthermore, the experimentally measured shift in V* occurred in a steep region of the activation curve for IK,dr, assuming typical parameters used to model Hodgkin-Huxley channels in hippocampal pyramidal neurons (Fig. 3C), and would therefore allow nonlinearly increased activation of IK,dr, consistent with the Morris-Lecar model reported in Fig. 2B. These data are therefore consistent with the postulated shift in balance of perithreshold currents that would account for the switch from class 1 to class 2 excitability.
Effects of shunting and adaptation on shape of the f-I curve
We then sought more direct evidence for a functional switch in excitability caused by shunting and adaptation. Because Hodgkin's original basis for classification was the shape of the f-I curve (see introduction), we tested whether shunting and adaptation changed the f-I curve from continuous to discontinuous, consistent with a switch in excitability from class 1 to class 2.
Shunting had two effects on the f-I curve according to simulations: it shifted the curve rightward and increased the minimum sustainable firing rate (fmin) to >0 spike/s, thus making the curve discontinuous (Fig. 4 A). To determine effects of shunting on fmin in experimental data while minimizing the influence of adaptation, we measured instantaneous firing rate from the reciprocal of the first interspike interval or ISI and compared f-I curves in the low- and high-conductance states. Like in simulations, shunting shifted experimental f-I curves rightward and increased fmin (Fig. 4B). Unlike in simulations, it is impossible to demonstrate experimentally that the neuron can spike at rates arbitrarily close to 0 spike/s (given the residual noise from ion channels, etc.), but, across all neurons tested, fmin increased significantly from 3.5 ± 1.0 spike/s under control conditions to 16.4 ± 1.4 spike/s when a 10-nS shunt was applied to the neuron (P < 0.001, t-test, n = 6 cells; Fig. 4C).
To determine if adaptation caused a similar increase in fmin, we measured instantaneous firing rate from the reciprocal of the last ISI within the initial burst (i.e., after development of adaptation). Once adapted, the same cell illustrated in Fig. 4B could not sustain spiking below ∼20 spike/s regardless of Istim (Fig. 4D). Across all neurons, adaptation significantly increased fmin in both the low- and high-conductance states to 16.2 ± 1.6 and 25.5 ± 1.6 spike/s, respectively (P < 0.005; Fig. 4E).
Thus shunting and tonic depolarization (which elicits adaptation) comparable to that experienced by neurons in vivo caused neuronal f-I curves to become discontinuous, consistent with the neuron switching from class 1 to class 2 excitability. By causing a switch to class 2 excitability, we predicted that shunting and adaptation encourage neurons to oscillate or resonate (i.e., respond more strongly to certain input frequencies), whereas those same neurons would not oscillate under control conditions when they are class 1 excitable.
Effects of shunting and adaptation on MPOs
We begin here by explaining the dynamical basis for the preceding prediction, i.e., why class 2 neurons oscillate whereas class 1 neurons do not. Class 1 neurons initiate their spikes on the basis of subthreshold inward current causing a SNIC bifurcation. This is reflected in the nonmonotonic steady-state I-V curve (Fig. 1Ca) and can be summarized by local stability analysis at the fixed point (see methods), which shows that ∂Iss/∂V = 0 at the saddle-node (Fig. 5Aa ). The implications are that fast-activating inward current (INa) does not compete with slow-activating outward current (IK,dr or IM) because the latter only starts to activate at suprathreshold voltages—this is why steady-state current is inward at perithreshold potentials and why the corresponding I-V curve has a region of negative slope. In class 2 neurons, on the other hand, spike initiation occurs through a Hopf bifurcation because net current is outward at threshold, as reflected in the monotonic steady-state I-V curve (Fig. 1C, b–d). This means that INa must compete with IK,dr (or IM) in a time-dependent manner. Subthreshold MPOs occur when inward current starts to activate but outward current catches up before a full spike occurs; this trajectory is manifested on the phase plane as spiraling around a stable focus ( at the fixed point; Fig. 5A, b and c) (see also Hutcheon and Yarom 2000). Repetitive spiking occurs when that focus becomes unstable via a subcritical Hopf bifurcation ().
By relating the generation of MPOs to the nullcline intersection, we can predict how conditions impact MPO frequency. MPO frequency reflects the rate of spiraling around the focus, which is reflected in the value of the complex component of the eigenvalue found by local stability analysis (Kaplan and Glass 1995). Equivalently, MPO frequency can be estimated from the phase plane: oscillations cannot occur when the V- and w-nullclines intersect tangentially, but MPO frequency increases as the nullcline intersection becomes less acute (i.e., as the angle becomes less sharp) (Fig. 5A, middle row). The preceding is true for a given set of rate constants, but MPO frequency also depends on how rapidly outward current activates (controlled by φw or τw) relative to instantaneous activation of inward current. Importantly, MPO frequency and fmin are directly related insofar as MPO frequency is less than fmin + Δf (Fig. 5A, bottom row), where Δf accounts for subthreshold MPOs occurring in the bistable region of the bifurcation diagram (Fig. 5B).
Thus based on phase plane geometry, we know that class 2 neurons should oscillate whereas class 1 neurons should not. Indeed the model did not exhibit subthreshold MPOs when depolarized to within a few millivolts of threshold under control conditions, when excitability was class 1 (Fig. 6 A, black), but subthreshold MPOs were observed after the model was converted to class 2 excitability (Fig. 6A, gray). To show that MPOs resulted from the switch in excitability rather than being a specific consequence of shunting or adaptation, the model shown in Fig. 6A was converted to class 2 excitability by changing an unrelated parameter, βm, which caused a depolarizing shift in the voltage-dependent activation curve for INa. Because greater depolarization was required to activate INa, this parameter change caused a depolarizing shift in V* (like in Fig. 2A); as a result, spikes were generated through a Hopf bifurcation (bifurcation diagram not shown). Reducing the maximal sodium conductance (ḡNa) has a comparable effect (see following text).
If, instead of adjusting βm, we added adaptation or shunting to the control model, the model exhibited subthreshold MPOs as predicted by its conversion to class 2 excitability (Fig. 6B). Furthermore, as predicted by local stability analysis (see corresponding phase planes in Fig. 5), the shunted neuron exhibited higher frequency MPOs than the adapted neuron given the parameters used (Fig. 6B). Also the amplitude (power) of MPOs was substantially less in the shunted neuron because low input resistance in the shunted neuron mitigated voltage changes. By stimulating our model with periodic inputs with different frequencies, we confirmed that class 2 models resonated at frequencies near their oscillation frequency, whereas the class 1 (control) model did not resonate, which is consistent with its inability to oscillate (Fig. 6C); both results are consistent with previous work (e.g., Izhikevich 2007).
We then compared these results with experimental data. Our recordings revealed subthreshold MPOs when CA1 pyramidal neurons were tonically depolarized. Oscillations were not necessarily obvious in the raw data, but power spectra showed a clear peak in the θ frequency range (Fig. 7 A). In the low-conductance state, frequency at the peak of the power spectrum corresponded to the interval between doublet spikes in the sample traces. In the high-conductance state, amplitude of MPOs was reduced (i.e., power was reduced and spiking was virtually abolished), but there was still a clear peak in the power spectrum, which was shifted to a higher frequency than in the low-conductance state (Fig. 7A). Changes in MPO frequency and amplitude were consistent with the effects of shunting observed in simulations (see Fig. 6B). Similarly, MPO frequency increased as Istim was increased (Fig. 7B), consistent with stronger stimulation eliciting stronger adaptation (i.e., larger outward current), which is again consistent with the model (simulation data not shown). As in our model, we confirmed that oscillating CA1 pyramidal neurons also resonated at θ frequency when stimulated with periodic input (Fig. 7C).
To determine whether CA1 pyramidal neurons could resonate without being tonically depolarized, we stimulated them with noisy current containing frequencies spanning the θ range (Fig. 7D, inset). This stimulation causes intermittent near-threshold depolarizations and can be used to reveal resonance the same way as periodic input (Schreiber et al. 2004). Using this protocol, we determined that CA1 pyramidal neurons in the low-conductance state that were not tonically depolarized did not resonate (Fig. 7D, black). However, when the same neuron was shunted, power was reduced at low frequencies and increased at high frequencies compared with the unshunted condition, which is consistent with the change in passive membrane properties (i.e., reduced input resistance and membrane time constant) but, more important here, the neuron exhibited resonance in the θ frequency range (Fig. 7D, red). Experimental data are therefore consistent with shunting and/or tonic depolarization converting CA1 pyramidal neurons from integrators to resonators.
Frequency of membrane potential oscillations
CA1 pyramidal neurons clearly oscillated in the θ frequency range (Fig. 7); however, our model neuron oscillated at significantly higher frequencies (Fig. 6). We therefore asked whether our model could produce lower frequency MPOs comparable to those observed experimentally. By reducing φw (which controls how rapidly gK,dr activates; see Eq. 2), MPO frequency was shifted down to the θ frequency range (Fig. 8A ). However, reducing φw enough to produce θ frequency MPOs caused spikes to become unphysiologically wide (Fig. 8A, inset). This suggests that the process controlling θ frequency MPOs is significantly slower than the process controlling spike repolarization, and that the model must therefore include a third time scale. As alluded to in methods, IM operates on a slower time scale than IK,dr; hence, we attempted to generate θ frequency MPOs in our 3D model by adjusting the parameters controlling IM, most notably τz. As predicted, appropriately tuned IM parameters resulted in θ frequency MPOs (Fig. 8B).
Time series in Fig. 8C illustrate how MPOs arise from the competition between inward and outward currents. During high-frequency MPOs, initial depolarization (which causes instantaneous activation of gNa) is followed after a short delay by activation of gK,dr, which causes hyperpolarization (Fig. 8Ca); although tonically activated, gM does not vary on the time scale of these fast oscillations—this was the basis for approximating IM as a constant for earlier analysis (see methods). During low-frequency MPOs, on the other hand, initial slow depolarization is followed with virtually no delay by activation of gK,dr, which means gK,dr does not provide the delayed outward current responsible for generating the oscillatory voltage change; instead, initial depolarization is followed by delayed activation of gM. In other words, slow voltage change allows enough time for gM to vary, despite its slow kinetics, and thus to participate directly in generating MPOs. This has an important technical implication insofar as IM cannot be approximated as remaining constant when explaining θ frequency MPOs. Nonetheless, generation of θ frequency MPOs can still be treated as a 2D problem (rather than a 3D one) because during slow oscillations, gNa activates slowly and with kinetics similar to gK,dr (see Fig. 8Cb); accordingly, IK,dr can be approximated as an instantaneously varying offset in the competition between slow-activating INa and ultraslow-activating IM.
The dynamics described above are also evident from phase plane analysis. Recall that w and z control activation of IK,dr and IM, respectively. In the 2D model without IM, shunting converted the intersection between the V- and w-nullclines from a saddle-node to a focus (Fig. 8Da), which is consistent with increased perithreshold activation of IK,dr (see Fig. 2); high-frequency MPOs result from spiraling around that focus (see Fig. 5). In the 3D model with IM, the V- and w-nullclines intersected at a saddle-node despite adaptation (hence, no high-frequency MPOs), but the V- and z-nullclines intersected at a focus (Fig. 8Db), which is consistent with perithreshold activation of IM; θ frequency MPOs result from spiraling around this focus. Spiraling is slower on the V-z phase plane because of z's long time constant (see preceding text). These changes in phase plane geometry are paralleled by changes in the steady-state I-V curve insofar as perithrehsold activation of IK,dr or IM both cause that curve to become monotonic (Fig. 8E).
To summarize, MPOs are generated when inward current competes with slower-activating outward current. We originally attributed MPOs to the competition between INa and IK,dr, but subsequent analysis suggested that θ frequency MPOs arise from competition between INa and an outward current slower than IK,dr, namely IM. The competition between INa and IM is the same as that between INa and IK,dr: initial activation of inward current is followed by delayed activation of outward current—this is manifested as spiraling on the respective phase plane. Interestingly, our results show that IM can, theoretically, contribute to oscillations in different ways: by competing directly with INa as described above (thus producing θ frequency MPOs) and also by providing a sustained outward current that shifts threshold and thus modulates the competition between INa and IK,dr (thus producing higher frequency MPOs; see Fig. 3).
Our experimental data from CA1 pyramidal neurons argue that IM functioned uniquely through the first mechanism. First, power spectral analysis revealed MPOs in the θ frequency range but not at higher frequencies (see Fig. 7). Second, adaptation caused an increase in fmin to between 10 and 30 spike/s, depending on shunting (see Fig. 4); this is high enough to allow θ frequency MPOs (see Fig. 5) and is therefore consistent with fmin being determined by competition between INa and IM but it is lower than would be predicted if fmin was determined by competition between INa and IK,dr. By extension, we would therefore argue that CA1 pyramidal neurons convert from class 1 to class 2 excitability primarily because of the outward current contributed by IM rather than because of the outward current contributed by IK,dr as initially hypothesized in Fig. 3. Importantly, voltage dependency of IM is such that IM can be activated at subthreshold voltages; by extension, a shift in V* can significantly modulate the strength of perithreshold activation (Fig. 9A ), which has already been shown to have important functional consequences (Prescott et al. 2006).
Effects of voltage threshold (V*) on strength of IM
Variable activation of IM (based on modulation of V*) is important for explaining certain aspects of our experimental results. For one, CA1 pyramidal neurons spiked at low rates on initial depolarization from rest (Fig. 4, B and C); although IM is slow to activate, it should be able prevent spiking below θ frequency (see preceding text). These data therefore suggest that IM was not strongly activated on initial depolarization in nonshunted CA1 pyramidal neurons. This is consistent with the absence of resonance when nonshunted neurons were transiently depolarized to perithreshold voltages (Fig. 7D). However, nonshunted CA1 neurons clearly oscillated/resonated during sustained perithreshold depolarization (Fig. 7, A and C), which indicates that IM did eventually activate under those conditions. Delayed activation of IM is consistent with the observation that adaptation increased fmin to >θ frequency (Fig. 4, D and E). One likely explanation for delayed activation of IM is that an even slower process, such as cumulative sodium channel inactivation that develops on the time scale of seconds (Fleidervish and Gutnick 1996), allows for increased activation of IM when the neuron experiences sustained depolarization. Sodium channel inactivation was evident in our experimental data, for example, based on slow decrement in spike amplitude during sustained depolarization (Fig. 9B). Other results suggest that shunting allows for increased perithreshold activation of IM: transient depolarizations caused θ frequency resonance (Fig. 7D) and fmin was >θ frequency on initial depolarization (Fig. 4, B and C) in shunted neurons. These data therefore suggest that shunting and cumulative sodium channel inactivation (or some other very slow process triggered by sustained depolarization) allow for increased perithreshold activation of IM that causes net steady-state current to become outward, thus converting CA1 pyramidal neurons from class 1 to class 2 excitability and allowing them to oscillate at θ frequency.
To demonstrate the feasibility of this explanation, Fig. 9C shows how a model neuron with IM oscillated at θ frequency when shunted or when sodium current was partially inactivated, whereas the same neuron did not oscillate (because IM was not strongly activated below V*) under control conditions (i.e., no inactivation or shunting). Removing IM from the model neuron prevented sodium channel inactivation (Fig. 9C, green curve) or shunting (data not shown) from causing oscillations, thus confirming that these two forms of modulation affected θ frequency MPOs by modulating activation of IM rather than by modulating activation of IK,dr. Consistent with causing increased perithreshold activation of IM, shunting and sodium channel inactivation converted the intersection between the V- and z-nullclines from a saddle-node to a focus (Fig. 9D) and made the steady-state I-V curve monotonic (Fig. 9E).
In this study, we used computational modeling and experiments in which in vivo-like conditions were recreated in an acute slice preparation to investigate whether CA1 pyramidal neurons behave as integrators or resonators, i.e., exhibit class 1 or 2 excitability according to Hodgkin's (1948) classification scheme. Based on changes in voltage threshold, shape of the f-I curve, and ability to oscillate/resonate, our results demonstrate that although pyramidal neurons behave as integrators under typical in vitro conditions, those same neurons behave as resonators when shunted and/or tonically depolarized as is likely to occur in vivo (see introduction). The change in coding strategy is fully accounted for by the relative increase in outward current caused by increased activation of IM, and the effects of that outward current on spike initiation.
The switch in excitability is interesting for several reasons. First, spike-initiating dynamics are typically thought of as an intrinsic property of the neuron, but results here indicate that spike initiation can be influenced by extrinsic factors like synaptic input (see also Steriade 2001). Second, the spike-initiating mechanism is mutable rather than being a fixed property of the neuron as evidenced by its qualitative change from SNIC bifurcation to Hopf bifurcation. But despite the qualitative change in bifurcation mechanism, both fmin and MPO frequency grow continuously from 0 Hz as the neuron transitions from class 1 to class 2 excitability (i.e., as net current at perithreshold potentials becomes more strongly outward). In fact, fmin and MPO frequency are directly related insofar as MPO frequency is less than fmin + Δf, where Δf accounts for subthreshold MPOs occurring in the bistable region of the bifurcation diagram (see results). Consistent with this, θ frequency (3–10 Hz) is less than our estimated value of fmin under in vivo conditions (∼15–20 Hz). This relationship between fmin and MPO frequency seems not to be broadly appreciated, but it suggests, for instance, that interneurons oscillating at γ frequency (30–80 Hz) must have even higher fmin, which appears to be the case (Erisir et al. 1999; Tateno et al. 2004).
Several studies have documented the importance of intrinsic membrane properties for network synchronization (Di Garbo et al. 2007; Geisler et al. 2005; Pfeuty et al. 2003; Tateno and Robinson 2007). Pfeuty et al. (2003) specifically concluded that potassium currents promote synchrony, whereas persistent sodium current impedes synchrony, which is consistent with our results insofar as outward current encourages class 2 excitability, whereas inward current encourages class 1 excitability. Golomb et al. (2007) recently showed that increasing the sodium window current (which contributes inward current at perithreshold potentials) changed the bifurcation mechanism and prevented subthreshold voltage fluctuations, consistent with inward current switching the excitability of their model neuron to class 1. Hutcheon et al. (1996) also showed through modeling that shunting could encourage resonance, although they attributed that effect to a quantitative change in the membrane time constant rather than a qualitative change in spike-initiating mechanism. Our results are also consistent with work showing that adaptation mediated by IM encourages class 2 excitability (Ermentrout et al. 2001), whereas cholinergic agonists, which block IM, have the opposite effect (Stiefel et al. 2008). Moreover, these results are consistent with reported effects of acetylcholine on low-frequency oscillations during slow-wave sleep (Steriade 2004). At first glance, our results appear to contradict Fernandez and White (2008), who showed that MPOs in entorhinal stellate cells were strongly attenuated by increased membrane conductance; in fact, our results are consistent insofar as shunting also attenuated MPOs in CA1 pyramidal cells and abolished spiking associated with those MPOs (see Fig. 7A). One must consider that, to oscillate, CA1 pyramidal cells require a depolarizing shift in voltage threshold (caused by shunting, for example) to allow sufficient subthreshold activation of IM for the neuron to convert to class 2 excitability; entorhinal stellate cells, on the other hand, are likely class 2 under “control” conditions. Therefore although shunting is required for oscillations in the former cell type, it is not required for oscillations in the latter.
In the initial stages of this study, we hypothesized that shunting and adaptation (mediated by IM) allowed for increased activation of the delayed rectifier potassium current IK,dr and that increased perithreshold activation of IK,dr was responsible for making net steady-state current outward, thus causing excitability to become class 2. However, although increased perithreshold activation of IK,dr could theoretically contribute to the generation of high (e.g., γ) frequency oscillations, subsequent analysis (see Fig. 8) indicated that IK,dr did not have the kinetics required to generate low, θ frequency oscillations. This result is, perhaps, quite obvious. Nonetheless, results of that initial analysis provided the insight necessary to explain differential activation of IM depending on operating conditions. Perithreshold activation of IM can be increased by the same depolarizing shift in voltage threshold that could increase perithreshold activation of IK,dr. Perithreshold activation of outward currents like IM contribute to the shift in threshold, but, more important for understanding modulation of perithreshold activation of IM are those processes that act on a faster or slower time scale than IM. Shunting and cumulative sodium channel inactivation caused by tonic depolarization are two such processes that cause a depolarizing shift in voltage threshold. Both are liable to be prominent under in vivo conditions but are modest under in vitro conditions, meaning IM is liable to have a less pronounced effect in vitro than it would have in vivo. The result is that CA1 pyramidal neurons that function as integrators (class 1 excitable) in slice experiments may function as resonators (class 2 excitable) in the intact, awake brain. This is important given that neuron models are usually built based on parameters measured in vitro; such models may fail to reproduce, even qualitatively, important phenomena like oscillations and resonance. For one, the classic leaky integrate-and-fire model may not be ideal for modeling pyramidal cells, although variations such as the exponential integrate-and-fire model seem to capture some of the nonlinear dynamics at spike threshold (e.g., Fourcaud-Trocme et al. 2003).
To conclude, the dynamical mechanism underlying spike initiation can change qualitatively depending on conditions such as the level of background synaptic input. A switch in spike-initiating mechanism has several computationally important consequences and adds to a growing list of potential difficulties in extrapolating from in vitro data to explain neuronal operation in vivo. Nonetheless, in vivo-like conditions can be recreated in vitro and represent a useful approach for resolving potential discrepancies. These results remind us that vigilance is necessary when using a reductionist approach that relies on experimental conditions (e.g., brain slices) that do not fully replicate the complexities of the intact brain. Model building and carefully designed experiments are necessary to recognize and surmount these shortcomings.
This research was supported by funding from the Howard Hughes Medical Institute to T. J. Sejnowksi and the Natural Sciences and Engineering Research Council of Canada to Y. De Koninck. S. A. Prescott was a long-term fellow with the Human Frontier Science Program, and Y. De Koninck is a Chercheur National with the Fonds de la Recherche en Santé du Québec.
We thank D. Spencer for helpful comments on the manuscript.
Present address of S. Ratté: Dept. of Neurobiology, University of Pittsburgh, 200 Lothrop St., Pittsburgh, PA 15213.
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- Copyright © 2008 by the American Physiological Society