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Keywords:

  • Computer simulations;
  • Recurrent networks;
  • Dentate gyrus;
  • Carbamazepine;
  • Phenytoin;
  • Markov model;
  • Mossy fiber sprouting;
  • Sodium channels;
  • Epilepsy;
  • Seizures

Summary

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

Purpose

A common notion of the mechanism by which the antiepileptic drugs (AEDs) carbamazepine and phenytoin act is that they block sodium channels by binding preferentially to the inactivated state, thereby allowing normal neuronal firing while blocking ictal activity. However, these drugs have unpredictable efficacy and, in some cases, may exacerbate seizures. Previous studies have suggested that reducing sodium channel availability in the dentate gyrus (DG) paradoxically increases excitability. We used a biophysically detailed computer model of the DG to test the hypothesis that AEDs increase excitability by disproportionately reducing negative feedback mechanisms.

Methods

We built a Markov model of sodium channel gating that reproduces responses to voltage clamp experiments in the presence of carbamazepine and phenytoin. We incorporated this validated Markov model into a biophysically realistic computer model of DG neurons and networks. Simulated drug concentrations were similar to those measured in cerebral spinal fluid in medicated patients. Single neuron models were stimulated with current injections, and networks were stimulated with perforant path synaptic input. In the network model, environmental effects were studied by introducing mossy fiber sprouting.

Key Findings

As expected, drugs reduced sodium channel availability, which in turn reduced action potential amplitude. This had only a small effect on action potential (AP) firing rate during brief (100 msec) current injections. Paradoxically, long current injections (2,500 msec) increased AP firing rates. This was caused by reduced calcium entry and consequently reduced activation of calcium activated potassium channels. It is important to note that the main determinant of drug effect was resting membrane potential (RMP) and not action potential firing rate. Binding of phenytoin and carbamazepine is slow and, thus drug effects are largely determined by the long term state of the RMP. This paradoxical AP firing increase was dependent on the unusually large calcium-activated potassium conductances expressed by DG granule cells. This predicts that drug efficacy in a given network will depend on the precise makeup of conductances in the network. RMP is expected to vary with the level of activity in the network. We simulated the effects of drugs on single shot stimulus responses in networks with mossy fiber sprouting and varied the RMP in all neurons as a model for network activity. For an RMP of −50 mV, representing an active network, drugs had no effect, or in some cases, increased excitability. Drugs had an increasingly larger inhibitory effect on network responses as RMP decreased. An important prediction is that drugs will be unable to block ictal activity invading an active network.

Significance

Our key findings are that drug effects depend on both intrinsic properties of the network and its behavioral state. This may explain the paradoxical and unpredictable effects of some AEDs on seizure control in some patients.

A major burden of epilepsy is its inherent unpredictability, not only for seizures and lifetime course of the disease but also because the unpredictability in the effectiveness of antiepileptic drugs (AEDs). Patients often trial multiple drug combinations over many months or years and still fail to achieve full relief from seizures. Epilepsy is sometimes considered a disease of neuronal hyperexcitability, and many AEDs seem to work by decreasing neuronal excitability. Yet AEDs that are effective in some patients may be ineffective or indeed exacerbate seizures in others. In this article, we look specifically at carbamazepine and phenytoin. Both drugs are used primarily for management of complex partial seizures and less commonly for other seizure types. Both drugs reliably exacerbate absence seizures (Lerman, 1986; Perucca et al., 1998; Liu et al., 2006) and can also increase seizure frequency and type in other generalized epilepsy syndromes. In corticohippocampal slices from postnatal day 7 rats, in which seizure-like activity was induced by low magnesium, carbamazepine effects were highly labile. In some experiments, the drug had no effect, whereas in others the patterns of seizures were altered with tonic-like phases decreasing but with an increase in interictal discharges (Quilichini et al., 2003). In the same preparation, phenytoin was mostly without effect. In organotypic hippocampal slices, both carbamazepine and phenytoin also produced complex changes in seizure-like activity. The duration of tonic- and clonic-like events decreased but their frequency increased (Quilichini et al., 2003). These data demonstrate that effects of AEDs are complex and cannot be predicted by application of the simple idea that they reduce neuronal excitability. As we have shown when modeling the effects of epilepsy mutations (Thomas et al., 2009, 2010), there are multiple levels of positive and negative feedback that can be potentially modulated, and the net result on neuron and network activity cannot be predicted by intuitive arguments.

Sodium channels are critical for action potential (AP) initiation and propagation; they are expressed in higher quantities than other channel types, and mutations cause epilepsy (Escayg & Goldin, 2010). As a consequence, drugs that target sodium channels are among the most common AEDs prescribed. Carbamazepine and phenytoin, along with oxcarbazepine, eslicarbazepine, lamotrigine, valproate, zonisamide, furosemide, and lacosamide, all block sodium channels as part of their proposed mechanism of action. Most drugs seem to prolong recovery from inactivation (Kuo & Bean, 1994; Kuo et al., 1997). Definitive experiments studying the effects of carbamazepine and phenytoin in hippocampal neurons have shown that these drugs bind infrequently to channels at membrane potentials typical of resting neurons and bind frequently when neurons are depolarized (Kuo & Bean, 1994; Kuo et al., 1997). Block develops slowly compared to open times suggesting that the drugs bind preferentially to an inactivated state. Furthermore, the kinetics are distinguishably faster than slow inactivation suggesting that the drugs bind specifically to the fast inactivated state. Both carbamazepine and phenytoin have qualitatively similar effects on sodium channels but are quantitatively different.

Although the paradoxical nature of AEDs is well recognized, hypotheses as to why there is such patient to patient and seizure to seizure variability in the same patient are lacking. This variability will be due in part to the genetic makeup of the patient causing differences in ion channel responses to voltage and drugs. In this study, we explored nongenetic mechanisms that may cause paradoxical variability. Neuronal networks are complex, nonlinear dynamic systems with multiple levels of positive and negative feedback. Computer simulation provides a way to predict the consequences of manipulating aspects of these interactions (Lytton, 2008; Thomas & Petrou, 2008). Most biophysically realistic neuron and network models are based on the Hodgkin-Huxley formalism for describing ion channel kinetics (Hodgkin & Huxley, 1952). An assumption of these models is that activation and inactivation are independent processes. Although activation and inactivation are not independent, Hodgkin-Huxley models are able reproduce standard voltage clamp experimental data with high fidelity (Thomas et al., 2007). However, because carbamazepine and phenytoin bind exclusively to the inactivated state, independence of activation and inactivation can no longer be assumed. We developed a method to extend standard Hodgkin-Huxley models to incorporate state-dependent drug binding that can accurately reproduce published data. Because these drugs are used primarily in partial epilepsy, we incorporated these extended ion channel models into a previously published model of dentate gyrus (DG) neurons and networks (Santhakumar et al., 2005; Dyhrfjeld-Johnsen et al., 2007; Morgan & Soltesz, 2008; Thomas et al., 2009, 2010). We also introduced mossy fiber sprouting into the simulations as a model of environmental interaction.

Methods

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

Sodium channel model

To predict the effects of state-dependent drug binding to sodium channels we developed a mathematical model combining both drug and voltage gating. Standard Hodgkin-Huxley models assume a number of gates that open and close independently of each other. The opening probability of activation gates increase as voltage increases, whereas the opening probability of the inactivation gate decreases with increasing voltage. This formalism cannot be extended to state-dependent drug binding by simply adding another gate, which closes when the drug is bound, because this will not distinguish block based on channel state.

Each Hodgkin-Huxley gate transitions randomly, and without memory of previous states, between open and closed states with voltage-dependent rates αx and βx where x indicates the gate type, m for activation, and h for inactivation.

  • display math(1)

A typical Hodgkin-Huxley sodium channel model has three activation gates and one inactivation gate and thus there are 24 = 16 possible combinations of gate states. However, the states in which the same number of activation gates are open are identical so this reduces to eight independent states. This model can be represented by the kinetic scheme shown in Figure 1A (Patlak, 1991; Johnston & Wu, 1994; Hille, 2001). Horizontal transitions from left to right indicate successive activation gates opening, and top to bottom transitions indicate the inactivation gate closing. The channel conducts only when all gates are open. The rate functions are the Hodgkin-Huxley rate functions so any Hodgkin-Huxley model can be converted into this mathematically equivalent form.

image

Figure 1. Kinetic models of sodium channel gating. (A) A kinetic scheme that is equivalent to a Hodgkin-Huxley model with three activation gates and one inactivation gate. Cx represents closed states, but available states corresponding to x activation gates being closed and the inactivation gate being open. Ix corresponds to states where the inactivation gate is closed. O is the open and conducting state corresponding to all gates being open. (B) IDx represents states in which the drug has bound. These states are available only through the inactivated state. In order to open, the channel must unbind from the drug.

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This model can now be extended by adding another row of transitions from the inactivated state to the inactivated and drug bound state (Fig. 1B). This enforces the requirement that a drug binds only to inactivated channels. The on rates, βD, are reported directly in the literature. The off rate, αD, is derived from the Kd as follows

  • display math(2)

The disassociation constants are also available from the literature. The on/off rates used in this study are shown in Table 1.

Table 1. Transition rates between drug-bound and unbound states
DrugαD (1/ms)βD (1/M/ms)References
Carbamazepine9.4 × 10−438Kuo et al. (1997)
Phenytoin7 × 10−510Kuo and Bean (1994)

Neuron and network models

We used a previously published and extensively studied model of the DG that contains morphologically realistic models of the predominant cell type, the granule cells, as well as other excitatory and inhibitory neurons (Santhakumar et al., 2005). The network model consisted of 500 granule cells, 6 basket cells, 15 mossy cells, and 6 hilar perforant-path associated (HIPP) cells. Granule and mossy cells are excitatory (α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid [AMPA] transmission), whereas basket and HIPP cells are inhibitory (γ-aminobutyric acid (GABA)A transmission). Neuron models had between 9 and 17 compartments describing the dendritic arbor and realistic conductances, including the fast sodium and potassium channels that directly form the AP, an A current, L, N, and T type calcium channels, hyperpolarization-activated cyclic nucleotide-gated (HCN) current, and slow voltage and calcium gated potassium channels. Network connectivity is as follows. Each HIPP cell contacts 160 granule cells, 4 mossy cells, and 4 basket cells. Each basket cell contacts 100 granule cells, 3 mossy cells, and 2 other basket cells. Each mossy cell contacts 200 granule cells, 3 other mossy cells, 1 basket cell, and 2 HIPP cells. In the absence of mossy fiber sprouting, granule cells contact one mossy, one basket cell and three HIPP cells. In networks with mossy fiber sprouting, granule cells make spatially localized contacts with other granule cells. The degree of mossy fiber sprouting is quantified as a percentage, with 100% corresponding to 100 connections from each granule to other granule cells. Mossy fiber connections were generated randomly for each run.

Our model was mathematically identical to model number 51781 available from ModelDB (http://senselab.med.yale.edu/modeldb/), with the exception of the sodium channel model.

The original sodium channel models, for all neuron types, were reimplemented as general Markov kinetic schemes with additional drug-bound states as described above. The model was instrumented so that drug type and concentration could be passed as parameters at run time. Network structure is generated randomly at run time. To ensure that results are the not an artifact of a particular network, all experiments presented herein were performed five times. Models used for voltage clamp experiments were implemented in Matlab (Mathworks, Natick, MA, U.S.A.). Neuron and network models were simulated using NEURON (Carnevale & Hines, 2009), and source code is available from ModelDB (http://senselab.med.yale.edu/modeldb/).

Results

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

Model validation

We tested the combined drug and voltage gating sodium channel model against published voltage clamp data. Experimental data were taken from acutely disassociated hippocampal neurons where sodium currents are likely to come from a mixture of unknown channel subunits, whereas the base model voltage gated sodium channel we used was derived from fitting data from NaV1.2 subunits (Thomas et al., 2007). To test the effect of drugs, the following experimental protocols were used (Kuo & Bean, 1994; Kuo et al., 1997). Neurons were held at −70 mV for at least 1 min in phenytoin concentrations of 0, 10, 30, or 100 μm and channel availability was tested by stepping the voltage to −10 mV. Comparison of this protocol between model and previously published experimental data (Kuo & Bean, 1994) is shown in Figure 2A. Because phenytoin binds preferentially to the inactivated state and holds the channel in the inactivated state, it gives the appearance of left shifting the voltage dependence of inactivation. To measure the apparent shift in the voltage dependence of inactivation, voltage was held at −100 mV, stepped to a test potential for 16 s, and availability was measured by a step to 0 mV. The shift in the V1/2 of the steady state availability curve for model and experiment are shown in Figure 2B. Similar experimental results have been reported for carbamazepine.

image

Figure 2. Comparison of the sodium channel model and experiment in the presence of phenytoin for two protocols. Data measured in rat hippocampal neurons is shown in the left panels and model responses are shown in the right hand panels. (A) Current responses from experimental neurons held in drug solution at −70 mV for 1 min and stepped to −10 mV. (B) Shifts in the V1/2 of the steady state availability curve for 16 s test potentials. The left hand plots are reproduced from Kuo and Bean (1994) with permission of the publisher.

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Granule cell responses

We tested the effects of drugs on the responses of DG granule cells to current injections. Drug concentrations were 20 μm for carbamazepine and 10 μm for phenytoin, as these are typical concentrations found in cerebral spinal fluid at therapeutic doses (Sherwin et al., 1973; Strandjord & Johannessen, 1980). We also varied the resting membrane potential for these neurons by varying reversal potential for the leak conductance, and neurons were initialized to equilibrium levels of bound drug. Voltage-dependent conductances are largely deactivated at these potentials so there was no substantial difference in resting membrane conductance. Results are show in Figure 3.

image

Figure 3. Response to dentate gyrus granule cells to 250 pA current injections for different drug solutions and resting membrane potentials. The resting membrane potentials for each column are −70, −60, and −50 mV, respectively. The top row shows the full response, the second row shows the first 100 msec of the response, and the last row shows the last 100 msec of the current injection.

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At −70 mV, the first APs are identical in control and drug-bound neurons. However, the firing frequency of drug-bound neurons is higher than control, whereas the AP amplitude is reduced. As expected, the reduced amplitude is due to reduced sodium channel availability. However, this reduces calcium entry, which in turn means that drug-bound neurons have smaller calcium-activated potassium conductances resulting in a higher AP firing frequency (Thomas et al., 2010). After several hundred milliseconds, AP firing ceases due to a combination of opening of calcium-activated potassium channels and inactivation of voltage gated calcium channels. As internal calcium concentrations return to rest, potassium channels close and firing resumes (see Fig. 2 in Thomas et al. (2010)). The return to high frequency firing is caused by calcium channel inactivation, and so neurons will remain in this high firing rate state indefinitely. The precise nature of this pattern is modulated by drug binding. At higher resting potentials, initial action potentials occur with similar timing to control, but the firing frequency is higher for drug bound neurons while APs are smaller. As calcium channels rapidly inactivate at these higher membrane potentials there is little calcium entry and hence little accommodation in firing due to calcium-activated potassium channels.

Network responses

In the absence of mossy fiber sprouting, these networks lack sufficient recurrent connections to maintain firing after a stimulus has ceased (Santhakumar et al., 2005; Dudek & Sutula, 2007; Thomas et al., 2010). However, the DG may act as a gate to prevent ictal activity spreading into deeper structures in the hippocampus (Heinemann et al., 1992; Lothman et al., 1992; Stringer & Lothman, 1992). We tested the ability of carbamazepine and phenytoin to modulate transmission of activity through the DG by driving the perforant path with random, constant average frequency, trains of super-threshold synaptic events and observing average AP firing frequency as a function of time (Fig. 4). Resting membrane potential of all neuron types in the network was also varied as described previously. Because the depolarization is driven by excitatory synaptic activity, we would expect all neurons to be depolarized in an active network. The network activity showed strong accommodation due to calcium-activated potassium channels in granule cells as previously observed (Thomas et al., 2010). Firing rate was higher for more positive resting membrane potentials. However, drug solutions had no effect on responses.

image

Figure 4. Time plots of dentate gyrus network responses, without mossy fiber sprouting, to constant unsynchronized, 20 Hz input. The Y-axis is the average granule cell firing rate in 100 msec bins. Drug solutions and resting membrane potential were varied as indicated in the legend. (A) 20 Hz input, (B) 50 Hz input.

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Although, several changes are observed in the DG of patients with temporal lobe epilepsy, mossy fiber sprouting seems to have the greatest impact on DG excitability (Santhakumar et al., 2005). We tested the effects of drugs in networks with mossy fiber sprouting. Figure 5 shows data for 8% mossy fiber sprouting, where 100% sprouting is defined as 100 connections from each granule cell to other granule cells. This level of sprouting greatly enhances the excitability of this network without making it unstable and thus a seizure focus. Similar results are seen for other small values of sprouting. In this experiment, the stimulus is a single super-threshold input from the perforant path to a subset of neurons. Depending on the degree of mossy fiber sprouting, activity will be transmitted from the stimulus region by pathologic fibers and recurrent feedback may cause neurons to fire several APs (Santhakumar et al., 2005). We tested networks with neurons at a variety of resting potentials and with or without drug. At negative resting potentials, −70 mV, drugs were successful in reducing the both the duration of response and distance it spread from the stimulus region. At a resting membrane potential of −50 mV, both drugs slightly increased the duration of the response.

image

Figure 5. Effects of drug and membrane potential on responses in neurons with mossy fiber sprouting. (A) These are raster plots with time on the X-axis, and cell number for each granule cell in the network on the Y-axis, and where dot represents an action potential. These networks had 8% sprouting (Santhakumar et al., 2005). The stimulus is applied at T = 0 to the lowest numbered 100 granule cells. Initial resting membrane potential and drug solutions are as indicated. (B) Membrane potential traces from a granule cell in the stimulus region.

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Discussion

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

Because of their high expression and importance in AP generation, sodium channels are attractive drug targets for a range of diseases (Meisler & Kearney, 2005; Nardi et al., 2012). Effective drugs must be able to reduce symptoms while continuing to allow normal function. Ideally, drugs could be chosen based on an individual patient's symptoms and history. In the case of phenytoin, carbamazepine, and other AEDs it is notoriously difficult to predict whether a drug will be effective, produce unacceptable side-effects, or indeed exacerbate seizures. State-dependent sodium channel blockers like phenytoin and carbamazepine are thought be effective anticonvulsants because they inhibit high frequency AP firing while allowing low frequency firing to continue unimpeded (Ayala et al., 1977; McLean & Macdonald, 1983; Adler et al., 1986; Kuo & Bean, 1994; Kuo et al., 1997). However, the clinical observation that these drugs are only effective in some patients and the observation that they exacerbate seizures in some cases belies this simple view.

In order to examine how state-dependent sodium channel block impacts neuron and network behavior, we extended standard models of voltage gating to include drug gating. The overwhelming majority of voltage gating models are based on the Hodgkin-Huxley formalism because of the modest experimental data required to build them and their ability to reproduce experimental findings (Thomas et al., 2007). The technique we have developed here allows Hodgkin-Huxley models to incorporate drug-bound states and, at least in the case of phenytoin and carbamazepine, reproduce experimental data. These drugs bind preferentially to the fast-inactivated states rather than the closed state, providing an apparent voltage dependence. Unbinding is slow, holding the channel in the inactivated state and thus reducing channel availability. Binding is also slow and so occurs predominantly when the neuron is depolarized for many seconds. These two observations have to led to the view that these drugs will only block sustained high frequency firing. Because these drugs also bind to the fast-inactivated states in preference to slow-inactivated states we did not include slow inactivation in our model.

Our data predict that the effects of state-dependent block depend on more than just firing rate. In the case of a single neuron, responses to AP firing invoke negative feedback by opening voltage gated calcium channels, which increases intracellular calcium, and which in turn opens calcium activated potassium channels. We have found in the current (Fig. 3) and in a previous study (Thomas et al., 2010) that inhibiting sodium channels has a larger effect on AP firing through this mechanism than through a reduction in sodium channel availability. Furthermore, we have previously predicted that this increased firing leads to increased transmitter release, despite reduced AP amplitude, and hence increased excitation.

We also incorporated our modified sodium channel model into a DG network model and tested the effects of drug binding in normal networks and networks with pathologic recurrent excitatory connections. In normal networks, drugs had no effect. Synaptic inputs, even when driven at high frequency, differ from current injections in that the membrane spends more time near rest. This in turn reduces the time in the inactivated state, thereby reducing the opportunity for drug to bind. This was true even for networks in which the resting membrane potential was held at −50 mV.

Networks with mossy fiber sprouting, modeling epilepsy disease conditions, showed very different responses depending on network state. When neurons in the networks had resting membrane potentials of −70 mV, phenytoin and carbamazepine were able to reduce both the duration of the response to a brief stimulus and the extent to which it propagated through the network. On the other hand, in networks in which neurons were depolarized to −50 mV, phenytoin and carbamazepine slightly increased the duration of responses.

These data demonstrate that the effects of drugs are more complex than simply blocking high frequency firing. As we have discussed, by disrupting negative feedback, state-dependent blockers may increase firing. This is likely to be highly dependent on the neuron type. Neurons in the DG express large calcium-dependent potassium conductances (Staley et al., 1992; Dudek & Sutula, 2007; Howard et al., 2007), whereas in other networks these conductances do not play such a large role. Both the degree of recurrent feedback and resting membrane potential are important for determining the effects of drugs in these networks. Without recurrent excitatory connections, drugs had no effect, whereas the presence of recurrent excitatory drive depolarizes neurons sufficiently that drug block can occur and this drive is able to overcome the effects of calcium-activated potassium channels. The effects of resting membrane potential on network activity were also contradictory. The simple model predicts that increased resting membrane potentials increases drug binding and should decrease AP firing. This has potential consequences for how drugs can influence seizure activity. Drugs are more likely to be effective when epileptiform activity rises rapidly from quiescent network states or when development of the seizures requires pathologic activity to propagate through a quiescent network. These observations may account for clinical variability of drug efficacy.

Resting membrane potential is determined by network state (Destexhe et al., 2001, 2003; Fellous et al., 2003). Neurons in active networks undergo constant synaptic bombardment and as a result are significantly more depolarized than neurons in quiescent networks. The degree of activity in the network is, in turn, determined by behavioral state, preictal dynamic trajectory at the focus, and the nature of the networks that seizures must propagate through as they evolve. Our observations are consistent with drugs reducing seizures, for example in networks with low resting membrane potential, or having no effect and indeed exacerbating seizures in networks with high resting membrane potential. Taken together, understanding the effects of drugs depends on the precise networks involved both in initiation and spread and their dynamical history.

Acknowledgments

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

This study is supported by National Health and Medical Research Council of Australia (NMHRC) program grant 400121 to S.P. S.P. acknowledges support from an NMHRC fellowship (1005050). The Florey Neuroscience Institutes are supported by Victorian State Government infrastructure funds. E.T. acknowledges the disappointing and consistent refusal of the NHMRC to fund mathematical modeling in neuroscience as a research activity for at least the last 15 years.

Disclosure

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References

None of the authors have any conflict of interest to disclose. We confirm that we have read the Journal's position on issues involved in ethical publication and affirm that this report is consistent with those guidelines.

References

  1. Top of page
  2. Summary
  3. Methods
  4. Results
  5. Discussion
  6. Acknowledgments
  7. Disclosure
  8. References