The use of mathematical modeling and computer simulation in understanding seizures is increasingly recognized as a critical activity. It has been the subject of a recent high-profile review Citation[1], a book Citation[2] and a session at the 2007 Society for Neuroscience meeting. Although not initially focused on epilepsy, the Blue Brain Project Citation[101] is the most ambitious attempt to date at simulating the whole brain, and will, no doubt, have profound implications for epilepsy. Modeling is not one technique, but uses tools from mathematics, statistics and numerical analysis to answer questions at a variety of temporal and spatial scales, and with varying requirements for quantitative fidelity Citation[1]. At one end of the spectrum are techniques used to provide qualitative and high-level understanding about a system. These are sometimes called ‘lumped models’, in part because assumptions are used to strip the model of fine, low-level detail. These models are simple, with few parameters, which in turn means that the relationships between parts of the system can be easily understood. Frequently, mathematical tools from dynamical systems theory Citation[3] can be used to provide a complete understanding of the system’s behavior for all parameter values, describing what behaviors the system is capable of (e.g., quiescent states, oscillations or chaotic behavior), and how the system switches from one behavior to another. In the case of seizures, these models can provide a qualitative understanding of how seizure states relate to normal brain states, how they might occur, or even predict that seizures are inherently unpredictable.
In this editorial we will discuss the other end of the modeling spectrum: biophysically detailed ‘realistic’ models. In some cases, there are overt structural changes to brain tissue that would seem to explain epilepsy, for example, trauma, birth defects or tumors. However, in most cases the causal chain from molecular, through synaptic and cellular, to network hyperexcitability and hypersynchronicity is unknown. Typically, experimental techniques provide data for only a small number of variables at a time. For example, patch clamp electrophysiology might provide, at best, data from a small number of neurons and local field potentials at a few locations Citation[4]. Experiments that analyze the electrophysiological effects of ion channel mutations will not provide data about the cellular consequences of these mutations. The behavior of a biological system emerges from many complex nonlinear interactions, often working in opposing directions. It is not possible to conclude that because effect A creates hyperexcitability in one context that it will in another. To take a simple example, many mutations in the SCN1A sodium channel gene cause complete loss of surface expression of the protein Citation[5]; yet, these mutations cause seizures, which are an excess of network activity. The reason for this is that the predominantly affected cells are inhibitory neurons Citation[6]. Modeling can precisely predict the consequences of these nonlinear interactions. One goal of realistic modeling is to take observations of subtle changes in a protein or a cell and to understand the consequences in a network. For example, how does a 5 mV shift in a sodium channel activation curve lead to seizures? This is not a question that can be addressed in a lumped model, where subtle details have been deliberately abstracted. In realistic models, however, as much detail as possible is incorporated so that the effect of changes in these details can be understood.
How does this work in practice?
Something that is frequently not appreciated is the amount of data required to build high quality models. Perhaps the most detailed model that has been published is the neocortical column model of Traub et al.Citation[7]. The computer program that implements this model is available online from ModelDB, a database of computer models Citation[102]. The code consists of over 40,000 lines of Fortran, the majority of which specify parameters. Input data that are needed include: voltage and calcium dependence of activation and inactivation, and rates of activation and inactivation for 20 channel types; passive electrical properties and detailed morphology for 14 different neuron types; distribution of conductance density along the neuronal surface; numbers of neurons; probability of connection between neurons for each possible synapse type; the locations of synapses on the dendritic morphology; and kinetics and conductance densities for each type of synaptic receptor. These data are typically gathered from experiments across different species, brain regions, animal ages, preparations and laboratories. A high quality model of the sort outlined here would take 1–2 years to build and another year to explore, longer than many experimental studies. However, once built, the model can be relatively quick to update with new experimental data. An appropriate analogy is an experimental preparation, which might take several years to develop and validate, but can be used to answer many related questions.
How are model outputs interpreted?
There is no single ‘successful run’ of the model. Rather, by exploring a range of parameters and inputs a picture is built up of possible network behaviors or states. Examples include states with different input/output relationships, different levels of synchronization, high and low activity, and the ability to propagate activity at different rates and different distances. The model will show how the system switches from one state to another (e.g., going from a normal state to a seizure), and which are the most important parameters for modulating a given state.
Two epilepsy examples
Epilepsy-causing sodium channel mutations often change multiple biophysical properties of the channel. While some of these changes on their own would increase neuronal excitability and others would decrease excitability, it is difficult to predict the total effect on neuron firing Citation[8]. Spampanato et al. built mathematical models of wild-type and mutant sodium channel gating to predict how these mutations increase neuronal excitability Citation[9]. Owing to the typically small effect of nonlethal mutations, in a follow-up study we ensured that our models reproduced the voltage clamp data and, in particular, reproduced the difference between the wild-type and mutant forms seen experimentally Citation[8,10], and also predicted a change in neuronal excitability. In addition, by varying a single sodium channel biophysical property at a time, we demonstrated that neuron excitability is most sensitive to channel opening and provides a predictive tool to interpret future voltage clamp data from mutant channels.
An example of an open question in epilepsy that could be addressed by realistic modeling relates to work from our own laboratory on a familial childhood absence epilepsy mutation in the γ2 subunit of the GABAA receptor, which shows a 13% reduction in amplitude of phasic inhibitory transmission in layer 2/3 of the somatosensory cortex Citation[11]. The question remains as to whether this change is sufficient to cause absence seizures or whether there are other, possibly larger, changes that are responsible? The observed deficit is small, so it is necessary to build accurate cellular and synaptic transmission models that reflect the subtle changes in phasic inhibition to enable predictions about network behavior in epilepsy. To model this, it is not only necessary to fit synaptic responses, but to do so in morphologically reconstructed neurons to capture all of the parameters that determine synaptic responses. Once this is achieved, we can use this information to probe the sensitivity of our model to physiologically constrained changes in phasic inhibition. As we explore the model, we would determine whether the model is sensitive to phasic inhibition or other parameters and use this information to design directed experiments that could be tested in our animal models.
Financial & competing interests disclosure
The authors have no relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript. This includes employment, consultancies, honoraria, stock ownership or options, expert testimony, grants or patents received or pending, or royalties.
No writing assistance was utilized in the production of this manuscript.
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Websites
- Blue Brain Project http://bluebrain.epfl.ch
- Yale Senselab: ModelDB http://senselab.med.yale.edu/modeldb/default.asp