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Purpose: Idiopathic epilepsy is caused by the complex interaction of genetic and environmental factors. The purpose of this study was to use computational approaches to explore the interaction between changes in sodium channel availability caused by mutations and mossy fiber sprouting.
Methods: We used a previously published biophysically realistic computer model of dentate gyrus neurons and networks. A sensitivity analysis probed the effects of typical mutation-like changes in either single- or multiple-gating parameters. Isolated neuron models were stimulated with current injections, and networks were stimulated with perforant path synaptic input. The gene–environment interaction was studied by including mossy fiber sprouting into these networks.
Results: Single neuron responses to current injections were complex, with increased sodium channel availability paradoxically reducing firing rates. In the absence of mossy fiber sprouting, control networks showed strong accommodation supporting the role of the dentate gyrus as a gate. Availability changes alone were not able to drive the networks into ictal states, even though they reduced the effectiveness of the dentate gyrus gate. Interestingly, the presence of electrophysiologic changes substantially increased the ability of mossy fiber sprouting to induce ictal activity.
Conclusion: (1) Increased sodium channel availability does not necessarily lead to increased firing rates, (2) network excitability is most sensitive to changes in steady state activation of sodium channels, (3) mutation-induced changes in availability reduce the effectiveness of the dentate gyrus gate, and (4) mutations interact strongly with structural network changes to allow ictal-like activity in the dentate gyrus.
Hundreds of epilepsy-causing ion channel mutations have now been identified (Reid et al., 2009). Because the exact genetic cause is known in such cases, these discoveries provide an opportunity to study the epileptogenic process (Yu et al., 2006; Ogiwara et al., 2007; Tan et al., 2007; Chiu et al., 2008). However, linking the molecular lesion to cellular phenotype to network hyperexcitability to seizure is extremely difficult. Furthermore, most mutations have incomplete penetrance and phenotypic heterogeneity, indicating that other genes or environmental factors are involved in causing the condition in humans. An example of this complexity is mutations in SCN1B, which encodes a sodium channel auxiliary subunit. The predominant phenotype is generalized epilepsy with febrile seizures plus (GEFS+), a familial syndrome in which febrile seizures persist into later childhood or adolescence. Patients with mutations in SCN1B have a greater risk for developing temporal lobe epilepsy (TLE) than do patients with other GEFS+ mutations, suggesting a causative role (Scheffer et al., 2007). With current technology, the main way to understand how these mutations cause epilepsy is with the use of animal models. But even so, the experimental complexity is enormous, and additional approaches are needed to help develop directed hypotheses (Lytton, 2008; Thomas & Petrou, 2008). A complementary approach is to build biophysically realistic computational models that, because of their relative experimental tractability, can help generate experimentally testable hypotheses.
Ion channel mutations have their effects through a variety of mechanisms including complete loss of the protein, trafficking deficits, and electrophysiologic changes in the way channels respond to voltage or ligands (Reid et al., 2009). Electrophysiologic effects of mutations are usually studied by expressing the relevant ion channels in heterologous systems and stimulating them with fixed voltage steps (voltage clamp) or ligand application. These protocols typically reveal several changes in the channel’s availability at different voltages. We use “availability” to describe the pool of channels that are conducting or are ready to move into the conducting state by a depolarizing stimulus. Often these experiments reveal opposing changes in availability or changes that have opposing effects at different phases of the action potential (Xu et al., 2007). Because of the complex nonlinear nature of ion channel gating and the electrical interaction between other ion channels in the membrane, it is difficult to predict the resulting outcome on neuronal excitability. One way around this impasse is to build accurate mathematical models of ion channel gating, neurons, and networks, for wild type and mutant channels, which can be incorporated into computer simulations (Spampanato et al., 2004; Thomas et al., 2007).
In addition to predicting the effects of specific mutations, it is important to understand which particular changes are having the predominant effect. For example, are changes in sodium channel activation more important than changes in fast inactivation? In a previous study, we performed a “sensitivity analysis” for sodium channels in simple neuron models (Thomas et al., 2007). We found that shifts in the voltage dependence of activation had the most profound influence on excitability. Left shifts, which increase availability, dramatically increased firing rate and lowered firing threshold. The simple neuron models were less sensitive to shifts in the voltage dependence of fast inactivation, and less sensitive again to changes in gating rates.
Herein we extend these studies in a number of ways. Firstly, we wanted to extend these predictions to more realistic neuron models. The simple neuron models that we used previously did not have dendrites and had only the minimum number of conductances required to generate action potentials. Secondly, although seizure susceptibility must start with a cellular phenotype, seizures are manifestly network phenomena. We, therefore, performed a sensitivity analysis in more realistic models, specifically the neurons and networks of the dentate gyrus (DG).
We chose the DG for a number of reasons. First, some sodium channel mutations cause febrile seizures and in some cases these patients go on to develop TLE (Scheffer et al., 2007), which has hippocampal involvement. Second, in insult models such as kainic acid kindling or head trauma, the DG becomes hyperexcitable because of mossy fiber sprouting, and may be the seizure focus (Santhakumar et al., 2005). Using a previously published model of the DG (Santhakumar et al., 2005; Dyhrfjeld-Johnsen et al., 2007; Morgan & Soltesz, 2008), we analyzed the sensitivity of individual neurons and networks, to single changes in sodium channel gating parameters. We also simulated the effect of a specific sodium mutation in conjunction with mossy fiber sprouting that models environmental influences. The mutation we used was SCN1B(R85H) in the sodium channel β1 accessory subunit, which causes a left shift in the steady state voltage dependence of activation (increasing availability), slows entry into fast inactivation (increasing availability), and slows recovery from fast inactivation (decreasing availability) (Xu et al., 2007).
We used a previously published model of the DG (Santhakumar et al., 2005), which consisted of 500 granule cells, 6 basket cells, 15 mossy cells, and 6 hilar perforant-path (HIPP) associated cells. Granule cells and mossy cells are excitatory (alpha-amino-3-hydroxyl-5-methyl-4-isoxazole-propionate [AMPA]-mediated transmission), and basket and HIPP cells are inhibitory [γ-aminobutyric acid receptor A (GABA)A–mediated transmission). Neuron models had between 9 and 17 compartments describing the dendritic arbor and realistic conductances including the fast sodium and potassium channels, which directly form the action potential: an A current, L, N, and T type calcium channels, 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 (Fig. 1). The degree of mossy fiber sprouting is quantified as a percentage, with 100% corresponding to 100 connections from each granule to other granule cells (Santhakumar et al., 2005). Networks were initialized differentially depending on the amount of mossy fiber sprouting required.
The model was identical to model number 51781 available from ModelDB (http://senselab.med.yale.edu/modeldb/), but with the following modifications. Sodium channels were modified so that they could enter a slow inactivated state. Typically, the voltage dependence of slow inactivation is similar to that of fast inactivation but with a much slower time-course (Xu et al., 2007). Therefore, in this model we used the fast inactivation voltage-dependence curve to describe the voltage dependence of slow inactivation and voltage-dependent rates from our previous data (Thomas et al., 2007). To mimic the effect of epilepsy-causing mutations, sodium channel gating was modified by shifting the activation or inactivation curves (i.e., , where is the voltage of half activation or inactivation) or by altering gating rates up or down by a specified factor (i.e., , where is the voltage-dependent rate).
Models were simulated using our own software (http://www.evan-thomas.net/parplex). The numerical method for advancing the state variables for each neuron was based on the standard method developed for branched morphologic structures with local nonlinearities (Hines, 1984; Mascagni & Sherman, 1998). The time step was adapted by comparing the result for a single step of size h to two half steps of size h/2 (Press et al., 1988) to maintain a desired accuracy. Integrity of the network as a whole is maintained according to an algorithm described previously in Thomas (2000).
About 3,000 network runs were performed. Networks with mossy fiber sprouting took between 40 s and 6 min of real time to simulate 200 ms of network time, depending on the level of network activity, on a standard 2 GHz PC. Code is available from ModelDB (http://senselab.med.yale.edu/modeldb/) or by contacting the authors.
Presynaptic calcium entry assays were performed off line by using the simulated membrane potential to determine the total charge carried through N-type calcium channels. The assumption is that the more total calcium that enters a presynaptic terminal, the more transmitter that is released. This is calculated as
These calculations were done in Matlab.
Slow inactivation of sodium channels
Many mutations affect the voltage dependence or rates of slow inactivation of sodium channels (Spampanato et al., 2001; Xu et al., 2007), with unknown consequences. Furthermore, the role of slow inactivation in network and single-neuron excitability is not well understood. For these reasons we performed a baseline characterization of the effects of adding sodium channel slow inactivation into these models. We characterized the effect of slow inactivation by simulating the four neurons types of the DG network model with current injections. In the interests of brevity, only granule cell data are presented; however, the behavior of other cell types was qualitatively similar. Because of the presence of large calcium-dependent potassium currents, excitatory cells of the DG are strongly accommodating (Staley et al., 1992). For this reason we used very long current injections, 10 s, and measured firing frequency over 500 ms at the beginning and the end of the current injections.
Slow inactivation has almost no influence on firing frequency at the beginning of the current injection (Fig. 2A). This can also be seen in the example membrane traces in Fig. 2D. However, slow inactivation does strongly influence firing on longer time-courses. Figure 2B shows that for current injections of more than 150 pA, the neuron with slow inactivation fired at higher frequency. The reason was that slow inactivation reduced sodium channel availability, which reduced action potential amplitude (Fig. 2D). The reduced amplitude reduced calcium entry, which reduced activation of calcium dependent potassium current (Fig. 2E). The reduced potassium current allowed neurons with slow inactivation to fire at a higher frequency.
This raises the question of how to measure excitability. Even though the neuron with slow inactivation is firing at a higher frequency, the smaller action potential may allow less calcium entry into the presynaptic structure, and this may result in less transmitter release. We used the soma membrane potential waveform to calculate the relative amount of calcium that would enter through N-type channels at a presynaptic terminal. The results of these calculations are shown in Fig. 2(C). The calcium entry curves show the same qualitative differences as the firing frequency curves, that is, that neurons with slow inactivation are actually more excitable than those without.
The data in Fig. 2E show instantaneous calcium-activated potassium current caused by calcium entry through a mixture of N- and L-type channels. As has been proposed previously, a single large action potential may allow more calcium entry than a single small action potential (Howard et al., 2007), and this is reproduced by our data. However, total calcium entry over 500 ms, through a different mix of calcium channels is higher because of the higher firing rate.
Neuron and network sensitivity to gating changes
Sensitivity to parameters of activation
Single cell responses
To determine which gating parameters have the most influence on granule cell excitability we performed a sensitivity analysis similar to that performed on simple neurons (Thomas et al., 2007). We shifted the voltage dependence of activation by ±5 mV or altered the rate of activation by a factor of 1.2 and determined frequency at the beginning and end of 10 s current injections.
Summary data and sample traces for 200-pA injections are shown in Fig. 3. Right shifting the voltage dependence of activation reduced sodium channel availability to the point that neurons were effectively silenced. Left shifting the voltage dependence of activation had little effect on firing frequency at the beginning of the current injection (Fig. 3A). However, at the end of the current injection, neurons with altered activation fired at a lower frequency than control neurons, but their action potentials had larger amplitudes (Fig. 3B). This is similar to the counter-intuitive effect of slow inactivation, and the mechanism is the same. We also calculated calcium entry and found that this assay of excitability was in qualitative agreement with action potential frequency (not shown). Figure 3 shows that the firing pattern of control and mutant neurons was quite different. In the control neuron, firing adapted as calcium entry activated potassium channels, but the adaptation disappeared as calcium channels inactivated. Neurons with mutant sodium channels, by contrast, showed more regular firing throughout the current injection. Altering the rate of activation had very little effect on neuron output, as measured by firing frequency and calcium entry.
The natural inputs to neurons are synaptic events, not current injections, so we simulated the networks as described in the methods. The stimuli consisted of super-threshold synaptic inputs to all granule, mossy, and basket cells from the perforant path (Santhakumar et al., 2005). Two input patterns were used, asynchronous random inputs described by a Poisson process and synchronized inputs.
The exact nature of input into the DG in vivo is unknown, but it is reasonable to assume that under some circumstances synchronized input is likely to be important. Some data from simulations using 20-Hz synchronized input are shown in Fig. 4A. These are raster plots for the control network and networks in which the voltage dependence of activation has been shifted by ±5 mV. Output responses from the network are synchronized because the synaptic inputs are synchronized and super-threshold. In the control case, the first input pulses are transmitted through the network with high fidelity, but this is gated to a greater or lesser extent for subsequent pulses. When the voltage dependence of activation is right-shifted, the second pulse is completely gated. The network then somewhat erratically allows later pulses through. When the voltage dependence of activation is left shifted the network loses its ability to gate input pulses.
Figure 4B shows networks responses to 2, 5, and 20 Hz asynchronous inputs as parameters of activation are varied. In the control case, network output frequency is close to the input frequency at 2 Hz. At higher frequencies the network response accommodates and is only able to fire at a steady state of about 4 Hz after 500 ms of stimulation. When the sodium channel voltage dependence of activation was left shifted, the excitability of the network was increased by almost completely eliminating the ability of the network to accommodate. Right shifting the voltage dependence of activation decreased excitability such that the network was not able to sustain firing above 2 Hz. Altering the rates of activation had no effect on network responses. In all cases, network activity ceased at the termination of the perforant path input. Figure 4C shows membrane potential traces for a granule cell in a network receiving asynchronous inputs, showing that the membrane is hyperpolarized compared to the membrane responses to current injections.
Sensitivity to parameters of fast inactivation
Figure 5A shows that neuron responses to current injections were strongly affected by left shifting the voltage dependence of fast inactivation. Firing rate was higher at the beginning of the injection, but neurons were silent at the end of the injection. Figure 5B shows that when the voltage dependence of fast inactivation is left shifted, sodium channel availability rapidly decreases to the point where neurons are unable to fire. As with previous manipulations, the calcium entry assay shows that the increased firing rate does indicate increased transmitter output. Right shifting the voltage dependence of fast inactivation caused a moderate decrease in excitability in the early part of the injection and a more dramatic decrease in excitability at the end of the injection. Right shifting the voltage dependence of fast inactivation also altered the pattern of firing (Fig. 5B).
Increasing the rate of fast inactivation slightly increased output (as measured by firing rate and calcium entry), and decreasing the rate of fast inactivation slightly decreased output. Altering fast inactivation had no discernible effect on network responses to synchronous (Fig. 5C) or asynchronous (not shown) input.
Sensitivity to parameters of slow inactivation
Shifting the voltage dependence of slow inactivation by 5 mV or altering the rate of activation by a factor of 1.2 had no effect on single neuron or network responses.
Interaction between altered channel activation and mossy fiber sprouting
Mutations may make individuals more susceptible to environmental insults rather than being the direct cause of epilepsy. Insults in animal models generate mossy fiber sprouting of granule cells (de Lanerolle et al., 1989; Houser et al., 1990; Babb et al., 1991; Sutula & Hermann, 1999; Santhakumar et al., 2001), which may be able to cause seizure-like activity in the DG (Santhakumar et al., 2005). We used the simulation to explore the hypothesis that genetic variations and mossy fiber sprouting may interact to increase network excitability more than either effect alone. The network was the same as before except for additional connections between granules cells. The stimulus in this case was a single action potential input from the perforant path applied to the central 100 neurons. Data from these runs are presented as raster plots (Fig. 6). The left hand column corresponds to the control network. For low amounts of sprouting there is very little afterdischarge. As the amount of sprouting increases, the afterdischarge becomes larger and propagates further from the stimulus region (Santhakumar et al., 2005). With 20% sprouting, the stimulus triggers activity that propagates throughout the network and generates a discharge lasting longer than 60 ms.
We compared the control network to a network in which gating properties of sodium channels were altered in manner similar to that caused by mutations in the β1 subunit (Thomas et al., 2007; Xu et al., 2007), specifically SCN1B(R85H). The voltage dependence of activation was shifted by −4.2 mV and the rate of inactivation was increased by 10%. The mutation profoundly affects the sensitivity of the network to sprouting. Afterdischarges occur for smaller amounts of sprouting, and the ability of activity to propagate throughout the network also occurs at lower levels of sprouting. When the sprouting reaches 20%, the excitability of the network is able to overcome the accommodation in the neurons and activity is sustained for at least 20 min, the duration of the longest run performed.
The rightmost column of Fig. 6 shows a network in which only the voltage dependence of activation has been shifted by −4.2 mV, but the rate of inactivation has not been altered from control. The network shows very similar excitability to that of the SCN1B(R85H) mutation, indicating that the shift in activation is the more important change introduced by the mutation.
From a dynamical systems point of view, epilepsy is a disease of brain homeostasis characterized by a loss of stability. In normal circumstances, the brain actively maintains stability through innumerable complex feedback mechanisms. The recent discovery of single gene mutations allows us to identify causes of instability and gives us the opportunity to understand some of the mechanisms that may result in a seizure-prone network. However, linking an ion channel mutation to a seizure is a difficult task. Even when considering only electrophysiologic changes, most mutations affect several aspects of sodium channel gating, some which increase excitability and others which decrease excitability (Thomas et al., 2007). Furthermore, how these interact with other conductances in a complex neuron and how these in turn affect network performance are hard to predict based on voltage clamp data alone. We, therefore, explored the consequences of typical mutation changes in a previously published model of the hippocampal DG network (Santhakumar et al., 2005). We chose this network because of its potential role in TLE (Dudek & Sutula, 2007) and to attempt to understand why some sodium channel mutations may cause TLE (Scheffer et al., 2007) but other mutations do not (Wallace et al., 2001; Kananura et al., 2002).
Single cell electrophysiology
Single cell electrophysiology is the best established method for characterizing the intrinsic dynamics of neurons, and this has carried over as way of assessing excitability in simple, single neuron models (Spampanato et al., 2004; Thomas et al., 2007). In these previous studies, the purpose of the simple models was to understand the cumulative effect of several gating changes on excitability and so they only included sodium, delayed rectifier potassium and leak conductances. However, nearly all neurons possess a much larger complement of conductances, which can cause complex firing patterns such as accommodation or bursting behavior. We found that the effects of gating changes were different from those in simple neurons and not predictable from channel availability arguments based on voltage clamp data.
The first unexpected finding was that, based on firing rate alone, gating changes often had opposite effects at the beginning or end of long stimuli. For example, left shifting the voltage dependence of activation had very little effect on average firing rate and transmitter release [unlike simple neuron models (Thomas et al., 2007)] at the beginning of the stimulus, but decreased firing at the end of the stimulus. Similarly, left shifting the voltage dependence of fast inactivation increased firing rate and transmitter release at the beginning of the stimulus but sodium channels rapidly inactivated silencing neurons after a few hundred milliseconds. As well as effects on firing rate, gating changes also altered the pattern of firing. This has potential implications for network behaviors, which require closely synchronized firing and may alter either rate based or spike timing based coding.
Sodium channel slow inactivation and epilepsy
Many mutations alter slow inactivation of sodium channels (Reid et al., 2009), but our data consistently suggest that these changes are unlikely to be responsible for seizures. We found that both neurons and networks of the DG are completely insensitive to mutation-like changes in voltage dependence or rates of slow inactivation. Completely removing the ability for sodium channels to slowly inactivate had the paradoxical result of decreasing excitability in response to long duration stimuli. However, no naturally occurring mutation has been reported that prevents entry into this state.
The natural input of neurons is not current injections but synaptic input from other neurons in a network. As previously noted, the control DG network is unexcitable (Dudek & Sutula, 2007), because the predominant cell type, the granule cell, has large calcium dependent potassium currents, which cause these neurons to strongly accommodate. Furthermore, in normal circumstances the network has only sparse recurrent excitatory connections. This means that in response to constant asynchronous inputs, the network response also accommodates. In response to synchronous inputs, the network effectively gates inputs after the first few pulses. This agrees with experimental observations in which the DG needs to be strongly activated before activity will propagate from the entorhinal cortex into deeper structures (Heinemann et al., 1992; Lothman et al., 1992; Stringer & Lothman, 1992). This has lead to the concept of DG gating, the ability of the DG to block or regulate the transmission of ictal activity into deeper structures in the hippocampus.
Unlike the current injection experiments, changing sodium channel gating produced excitability changes in the DG network that might be predicted from channel availability arguments. For example, left-shifting the voltage dependence of activation increases sodium channel availability at any given membrane potential and so increases depolarizing drive. Networks with these changes showed reduced accommodation and reduced gating of synchronous inputs. However, once the input stopped, networks immediately returned to the quiescent state. Therefore, without additional structural changes in the DG, it is unlikely to be a seizure focus. However, if there is also increased excitability in deeper structures in the hippocampus caused by the mutation, then compromised DG gating may play an important role in allowing the hippocampus to be driven into seizure activity.
Another unexpected finding is that it is difficult to predict the effect of a mutation on the DG network based on changes in single neuron responses to current injections. Most changes in network responses were very small compared to changes in single neuron firing. This is because current injection responses are based on long trains of action potentials, whereas neuron in networks, even networks with mossy fiber sprouting, fire short trains or single action potentials. Another reason for this difference is that the irregular nature of synaptic input allows the cell to return to relatively hyperpolarized levels compared to current injections. At these membrane potentials, recovery from fast inactivation can be rapid and complete, reducing the impact of any changes to fast inactivation. Networks were sensitive to changes in voltage dependence of activation because this also affects action potential firing threshold. An alternative experimental technique that may address these issues is injection of synaptic conductance noise using dynamic clamp (Fellous et al., 2003), which may better mimic the high frequency synaptic bombardment thought to occur in vivo. Synaptic conductance noise can also be simulated in neuron models, potentially making this a better overall paradigm for comparing model predictions to experimental observations.
The causal relationship between mossy fiber sprouting in the DG and TLE is still debated (Dudek & Sutula, 2007). It is, therefore, important to understand the interaction between mossy fiber sprouting and ion channel gating changes. Left shifts in activation of sodium channels greatly exacerbated the effects of mossy fiber sprouting. The level of sprouting required for a self-sustaining wave of activity to propagate throughout the network was lower and the duration of the poststimulus was longer. At higher levels of sprouting, and with electrophysiologic changes similar to those caused by the SCN1B(R85H) mutation, the excitatory feedback in the network was such that the network entered a stable uncontrolled firing state. The worsening of the disease in TLE seems to be a positive feedback loop in which seizures drive further proepileptic changes in the limbic networks. Mutations, such as SCN1B(R85H), may act by significantly lowering the threshold for the epileptogenic process to start. This may be particularly relevant to the C121W mutation in the SCN1B gene that has been linked to increased susceptibility to TLE in patients harboring the mutation (Scheffer et al., 2007).
EAT was partially supported by an Australian National Health and Medical Research Council Peter Doherty Fellowship. CAR was supported by a University of Melbourne RD Wright Fellowship and by Australian National Health and Medical Research Council project grant 454655. Aspects of this work were also supported by an Australian National Health and Medical Research Council program grant 400121. 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.
Disclosure: None of the authors has any conflict of interest to disclose.