Exponential distributions are characterized by a single variable, the distribution mean. In this section, the dependence of the mean duration of normal and paroxysmal epochs on model parameters is analyzed. Other quantifiers such as total paroxysmal duration or paroxysm incidence can be derived from these two quantifiers. The quantifiers chosen in this study have straightforward interpretation. The first one tells how long one can expect a paroxysm to last on average and therefore it corresponds to the probability of termination of a paroxysmal epoch. The second quantifier tells, once a paroxysm has finished, how long we can expect to wait for the next one to occur and therefore it corresponds to probability of initiation of seizure-like activity.
For the analysis of the dependence of the model's behavior on parameters, we selected six out of 65 model parameters. We selected the parameters that are either assumed to play a role in the pathophysiology of absence seizures in animals and humans, or are assumed to be targets of antiepileptic drugs, or are associated with seizure activation methods (sleep, hyperventilation). We varied one parameter at a time while all others were kept constant. For each parameter setting we simulated 24 hours of activity and created duration histograms from detected paroxysmal and normal epochs. The histograms were fitted with exponential distributions and the distributions' means were calculated. Each parameter (except cholinergic modulation) was manipulated such that the system's behavior varied from at least one paroxysmal event during 24 hours of activity to a state of continuous paroxysmal activity. The influence of a cholinergic neuromodulatory input originating from the brainstem mesencephalic cholinergic neurons was investigated by applying additional DC offset simultaneously to membrane potential in the TC and RE populations. This offset was varied in the range —4 to 4mV in the TC and in the range 8mV to — 8mV in the RE population (+1mV shift in TC corresponded to —2mV shift in RE). This is justified taking into account that acetylcholine released by cholinergic pathways decreases a potassium conductance in the TC cells that brings about depolarization of the TC population, while it increases a potassium conductance in the RE neurons and thus induces hyperpolarization of the RE population (McCormick and Prince, 1986, 1987). Results of the analysis are summarized in Figure 25.5.
For each parameter, two panels are presented where the mean duration of paroxysmal epochs (left panel) and of normal epochs (right panel) are shown. The change of a parameter is given in percentage of the corresponding reference value. An increase of the duration of paroxysms and a decrease of the intervals between paroxysms can result from a series of factors: a reduction of cortical GABAa inhibition (see Figure 25.5A, left panel), reduction of intra-RE GABAa inhibition (see Figure 25.5E, left panel), withdrawal of thalamic cholinergic modulation (see Figure 25.5F, left panel), an increase of the slope of the sigmoid in the cortical interneuronal population (see Figure 25.5B, left panel), an increase of burst firing in the RE or TC populations (see Figure 25.5C, D, left panel). The mean paroxysmal epoch duration is always monotonic function of parameter change. In some plots, this dependence is linear on a logarithmic scale (see Figure 25.5C,E) indicating that, in these cases, the mean epochs duration has exponential dependence on the varied parameter. The local slope of the graph is related to the sensitivity of the system to a change of the given parameter. From Figure 25.5A,B, it follows that the system is most sensitive to cortical GABAA inhibition and to the slope of the sigmoid in the population of cortical interneurons, since in these graphs the operating range along the x-axes are the smallest while the range of the y-axes are comparable to the other graphs.
Another important parameter in the model is the noise level. We analyzed the model's performance by varying the variance of one noise source (cortical input PCx) while assuming that the other noise source (sensory input P) had a variance equal to zero. The results are summarized in Figure 25.6 where, in the left column, phase portraits of the system are presented and, in the right column, model outputs are shown. The phase portraits were created by plotting the mean membrane potential of the population of interneurons (VInt) versus that of pyramidal cells population (VCx). Three panels
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