In this section, we present the class of neural mass models we consider, which includes the following models that share the same mathematical structure in their dynamics:
(i) The model by Wendling et. al. in [155] that captures epileptic activity in the hippocampus, (ii) The model by Jansen and Rit in [74] on the generation of evoked potentials due to visual stimulation, and (iii) The model by Stam et. al. in [138] for the generation of alpha rhythms. These models have their origins in cortical column models by Wilson and Cowan [156], Freeman [50] and Lopes da Silva et. al. [99;100]. In the proceeding sections, the class of neural mass models are presented with decreasing level of complexity. The functional connections between the neural populations for each of the models and their corresponding more detailed block diagrams are shown in Figure 2.2-2.7.
2.3.1 Neural mass model by Wendling et. al.
Wendling et. al. built upon the Jansen and Rit model described in the Section 2.3.2.
Four neural populations (with one population being a subset of another) are included in this model as shown in Figure2.2.
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Figure 2.2: Functional relationship between neural populations for the model by Wendling et. al..
The populations are the pyramidal neurons, the excitatory population (included in the pyramidal neurons), the slow and fast inhibitory populations. The fast somatic projection of the inhibitory interneurons is introduced in this model because it is hy-pothesised to play a role in the fast oscillatory pattern seen in the EEG at the onset of an epileptic seizure. Wendling et. al. identified three parameters, namely the synaptic gains of the excitatory, slow inhibitory and fast inhibitory interneurons to result in the
2. A class of neural mass models
model producing EEG patterns that are known to be related to neurological events, from normal background activity to epileptic seizures. This provides great motivation for estimating these parameters.
Figure2.3describes the interaction between populations of neurons in greater detail, which consists of postsynaptic membrane potential (PSP) kernels he, hiand hg, sigmoid functions S : R→ R and connectivity constants C1 to C7.
Figure 2.3: Detailed block diagram of the Wendling et al. model. Reproduced from Figure 4 in [155].
The firing rate of the afferent population is converted into an excitatory, slow or fast inhibitory postsynaptic membrane potential via the following kernels, for t≥ 0:
• The excitatory population:
he(t) = θAat exp(−at). (2.1)
• The slow inhibitory population:
hi(t) = θBbt exp(−bt). (2.2)
• The fast inhibitory population:
hg(t) = θGgt exp(−gt). (2.3)
Parameters θA, θB and θGin (2.1)-(2.3) correspond to the synaptic gains of the excita-tory, slow and fast inhibitory populations respectively. These parameters characterise
2. A class of neural mass models
the observed pattern in the EEG. For example, the values of θA, θB and θG that dis-tinguish between seizure and non-seizure activities have been identified in [155].
Internal variables x11, . . . , x71are introduced as shown in Figure2.3. They describe the membrane potential contribution from one population to another. For example, referring to Figure 2.3, the mean membrane potential of the pyramidal neurons is x11−x21−x31, which reflects the membrane potential contribution from the excitatory, slow and fast inhibitory populations, respectively. The mean membrane potential of a population is converted into the average firing rate of all the neurons in that population using a sigmoid function S:
S(z) = α2
1 + exp − r2(z− V2) , for z∈ R, (2.4) where α2 is the maximum firing rate of the population, r2 is the slope of the sigmoid and V2 is the threshold of the population’s mean membrane potential.
The neural populations are connected with connectivity strengths C1 to C7, which represents the average number of synaptic contacts between the neural populations concerned.
2.3.2 Neural mass model by Jansen and Rit
The interactions between the pyramidal neurons, excitatory and inhibitory populations (Figure2.4) are described in this model to investigate the generation of evoked poten-tials in the cerebral cortex. A detailed block diagram is provided in Figure2.5. As this model was extended upon by Wendling et. al. (whose model is presented in Section 2.3.1), each component of the block diagram: the PSP kernels he and hi and sigmoidal function S are as described in Section2.3.1.
2.3.3 Neural mass model by Stam et. al.
The model by Stam et. al. includes an excitatory and inhibitory population (Figure2.6) to replicate the alpha rhythms in the EEG. This event is related to the human subject being in a relaxed state with the eyes closed. Hence, the estimation of the unmeasured postsynaptic potential (PSP) of neural populations may better our understanding of the visual pathway while in an idle state.
Figure 2.7 shows a detailed block diagram of the model. This model differs from the models by Wendling et. al. and Jansen and Rit in the sense that the firing rate of a population is converted to a postsynaptic potential via different kernels from (2.1)
2. A class of neural mass models
Figure 2.4: Functional relationship between neural populations for the model by Jansen and Rit.
Figure 2.5: Detailed block diagram of the Jansen and Rit model.
Excitatory*
Figure 2.6: Functional relationship between neural populations for the model by Stam et. al..
2. A class of neural mass models
Figure 2.7: Detailed block diagram of the Stam et. al. model.
and (2.2), for t≥ 0:
• Excitatory population:
he(t) = θA[exp(−a1t)− exp(−a2t)]. (2.5)
• Inhibitory population:
hi(t) = θB[exp(−b1t)− exp(−b2t)]. (2.6)
Also, the sigmoid function that converts the postsynaptic potential to the firing rate of the population differs from (2.4) for the models by Wendling et al. and Jansen and Rit, as follows: where α1 is the maximum firing rate of the population, r1 is the slope of the sigmoid and V1 is the threshold of the population’s mean membrane potential.