**2.2 Modelling postsynaptic potentials of neuronal networks**

The highest contribution to Local Field Potentials (LFP) and the Electroencephalogram (EEG) signal comes from the postsynaptic membrane potentials of pyramidal

#### **Figure 4.**

*Neurophysiological basis of the EEG potentials recorded on the scalp surface from the geometrical summation of minute current dipoles of millions of pyramidal cortical neurons with variable magnitude, polarity and orientation depending on the local dominant circuitries implicated each time. Reproduced with permission from Tatum et al. [30].*

cells of cortical layer V creating a local current sink or rise, equivalent to a minute electrical dipole pointing at variable directions, depending on the local geometry (infoldings with gyri and sulci) and local circuitry of the cortical region, cortical neurons implicated and their spatial relationship with the scalp sensor (**Figure 4**) [29–32]. In addition, there is volume conducted electrical activity from every other region of the brain picked up at all scalp sensors, which is difficult to disentangle from the particular region in question, especially when other sources dominate the background (known as the reverse problem of EEG source localisation). To make things worse, the scalp and all the layers of tissue that intervene between the cerebral cortex and the scalp surface sensors act as low-pass filters, filtering out most of the high-frequency (mostly gamma range) activities of the EEG and severely contaminating the recorded EEG with muscle activity, eye or movement and other artefacts. The extracranial EEG may have excellent temporal resolution (down to the millisecond scale), but has fairly poor spatial resolution [29, 30, 32].

In principle, for an electrical signal deflection to be detected on the scalp sensor as an electrical potential difference between the underlying cortical region and a less active reference electrode, activities of millions of cortical/ pyramidal neurons need to be spatiotemporally summated over an area of about 4–9 (2<sup>2</sup> –3<sup>2</sup> ) cm<sup>2</sup> of cerebral cortex [30, 32]. The larger the numbers of cortical/ pyramidal neurons and the more synchronous their depolarization (corresponding to firing rates or bursts of action potentials), the greater the deflection on the scalp EEG (if at right angles to the cortical surface and current lines are traveling in the same direction). Although there are asymmetries between the depolarizing and repolarizing regions of the pyramidal neurons as reflected in current lines and waveform morphologies, current lines are mainly created by the depolarizing wave front and fan out away from the current sink, reflecting the volume conductive properties of that brain region. In essence every depolarization wave-front creates a minute electrical dipole and depending on the direction of that (negative at the apical dendrites vs. positive at the somata or vice versa), the radial components of that to the cortical surface will constructively contribute towards a positive or respectively negative deflection on the scalp EEG signal (**Figure 4**) [29, 30, 32].

None of the shortfalls of scalp EEG is a limiting factor with the use of in-depth electrodes robotically implanted (with accurate coordinates) in the human brain for epilepsy surgery purposes, which can reach even the deepest regions of the brain and record intracranial stereo-EEG from about 1 mm<sup>3</sup> voxels of brain tissue (the issue is which specific points in the brain these multiple depth contacts should target) [29].

The hallmark of epilepsy on the EEG is the interictal epileptiform spike or sharp wave, excessive or hypersynchronous discharges and paroxysmal rhythmical activities or attenuation changes in the background EEG activity before, during and after the seizure. The paroxysmal or synchronous depolarisation of millions of cortical neurons (converted in bursts of spikes or tonic neuronal firing) allows for spatiotemporal summation of the electrical activity of millions of cortical neurons and underlies the excessive or hypersynchronous discharges and other epileptiform phenomena we detect on scalp EEG. The cellular neurophysiological correlate of the interictal and ictal epileptiform discharges is the paroxysmal depolarization shift (PDS) of individual cortical neurons. The PDS is a prolonged (calcium-dependent) neuronal depolarization that results in multiple sodiumchannel (inward) current-mediated action potentials during depolarization,

followed by a prominent after-hyperpolarization phase beyond the baseline resting membrane potential, mediated by calcium-dependent potassium-channel (outward) currents and/or gamma-aminobutyric acid (GABA)-activated chloride (Cl) currents [33–36].

#### **2.3 Modelling action potentials of neuronal networks**

Using biophysically accurate Hodgkin-Huxley-type models for action potentials of neuronal networks would be computationally so demanding [37] that we could simulate only few neurons in real time. Using an integrate-and-fire model is computationally very effective, but the model is unrealistically simple and incapable of producing the rich spiking and bursting dynamics exhibited by cortical neurons. On balance, we adopted the simple spiking model proposed by Izhikevich [38], which is as biologically plausible as the *Hodgkin-Huxley* model, yet as computationally efficient as the *integrate-and-fire* model.

The spiking model developed by Izhikevich [38] essentially represents the complex integration of depolarising forces (voltage-gated and leaky sodium channels/inward currents) and repolarising forces (voltage-gated and leaky potassium channels/outward currents) by determining few key parameters: (a) the rate of decay of the action potential, (b) the sensitivity threshold for triggering the action potential, (c) the reset level of the depolarisation potential of the membrane and (d) the reset level of the repolarisation potential of the membrane. Depending on these four parameters, the model reproduces spiking and bursting behavior of different known types of cortical neurons as shown in **Figure 5**, and allows for dynamic functional connectivity and emergence of waves and rhythms at different scales of neural organization [39].

#### **Figure 5.**

*Known types of cortical neurons correspond to different values of the parameters a, b, c, d in the model described by the equations in the left upper box. RS, IB, and CH are cortical excitatory neurons. FS and LTS are cortical inhibitory interneurons. Each inset shows a voltage response of the model neuron to a step of dc-current I = 10 (bottom). Time resolution is 0.1 ms. This figure has been reproduced with permission from Izhikevich [37–42].*

#### **2.4 Cortical neuronal spiking model**

Neocortical neurons in the mammalian brain are classified into several types according to the pattern of spiking and bursting seen in intracellular recordings [38]. **Excitatory** cortical cells are divided into 4 different classes, with *RS (regular spiking)* neurons being the most typical neurons of the cortex. Upon a prolonged stimulus (injected DC-current, equivalent in a particular neuron to spatial convergence and temporal summation of more EPSP vs. IPSP) the neurons fire a few spikes with short interspike period and then the period increases (*spike frequency adaptation*). Increasing the strength of the injected DC-current increases the *interspike frequency*, though this can never become too fast because of large spike-after hyperpolarizations. **Inhibitory** cortical cells are usually divided into two classes, the most common being the *FS* (*fast spiking*) neurons that can fire periodic trains of action potentials with extremely high frequency, practically without any adaptation (slowing down). The other type of cortical inhibitory interneurons is the *LTS* (*low threshold spiking*) neurons that start spiking when a minimum low-threshold has been reached [37–42].

The best way to simulate the different dynamics of different neurons, is to assign for each excitatory cell (ai, bi) = (0.02, 0.2) and (ci, di)=(65, 8) + (15, 6) ri 2 , where ri is a random variable uniformly distributed on the interval [0, 1], and i is the neuron index. Thus, ri = 0 corresponds to regular spiking (RS) cell, while ri = 1 corresponds to the chattering (CH) cell. We use ri <sup>2</sup> to bias the distribution toward RS cells. Similarly, each inhibitory cell has (ai, bi) = (0.02, 0.25) + (0.08, 0.05) ri and (ci, di)=(65, 2) [38].

In addition, the Izhikevich model [39] can reproduce the behavior of **thalamocortical neurons**, which provide the major input to the cortex. *TC (thalamo-cortical)* neurons have two firing regimes: When at rest (v is around 60 mV) and then get depolarized, they exhibit tonic firing. However, if a negative current step is delivered and the membrane potential gets hyperpolarized (v is around 90 mV), the neurons fire a rebound burst of action potentials (bursting firing). The dynamics of other neuronal types, including those in **hippocampus**, basal ganglia, brainstem, and olfactory bulb, can also be simulated by this model [41].

The proposed Izhikevich simulation model [42] belongs to the class of pulse-coupled neural networks (PCNN). The synaptic connection weights among the neurons are described by a matrix S = (sij). Firing of the jth neuron instantaneously changes the variable vi by sij [38]. Although the network is connected randomly and there is no synaptic plasticity included, the neurons tend to self-organize into assemblies that exhibit collective rhythmical behavior in a frequency range corresponding to that of the awake mammalian cortex. Changing the relative strength (weights) of synaptic connections (excitatory and inhibitory) and the strength of the thalamic drive can produce other types of collective behavior, including spindle waves and larger slow sleep oscillations [39]. Therefore, our spiking model is fairly adaptable and can reproduce the dynamics of many different known types of neocortical and allocortical neurons (biological plausibility), while its high computational efficiency allows us to observe and study collective cortical states at a global neuronal network level [37–42].

## **3. Epileptogenic brain networks**

#### **3.1 Cortico-thalamocortical neuronal networks**

The reticular nucleus of the thalamus is part of the ascending reticular activating system (RAS) that modulates thalamocortical pathways and helps to synchronize and *Clinical Neurophysiology of Epileptogenic Networks DOI: http://dx.doi.org/10.5772/intechopen.104952*

#### **Figure 6.**

*The thalamocortical network. Please note the projection of specific and association thalamic nuclei project to layers III and IV of the neocortex and the projection of nonspecific thalamic nuclei to more superficial layers as well, layer I and II, allowing a temporal binding and integrative processing of information via cortical coincidence detection of specific and nonspecific thalamocortical inputs. By Zacharybarry (talk)—self-made, Public Domain, https://en. wikipedia.org/w/index.php?curid=33934402.*

desynchronize the EEG. Typically, the EEG is synchronized when thalamic relay neurons are in burst mode, and desynchronized when they are in tonic mode. Stimulation of the RAS suppresses slower activity (e.g. delta or theta) and stimulates faster activity (e.g. beta and gamma) and vice versa. This modulation leads to remarkable changes in cerebral electrical activity during wakefulness and sleep [43].

Most regions of cortex receive converging inputs from both specific (or association) nuclei and nonspecific thalamic nuclei (**Figure 6**). Specific and association thalamic nuclei project most commonly to layers III and IV of the neocortex, whereas nonspecific thalamic nuclei may reach more superficial layers, including layer I and II. This arrangement allows for regions such as the reticular formation to alter cortical

excitability levels by modulating the responsiveness of cortical neurons to inputs from specific or association thalamic nuclei. Furthermore, specific/association and nonspecific thalamic nuclei share reciprocal connections with the cerebral cortex. These provide a feedback mechanism from the cortex to the thalamus, which serves to control the amount of input that reaches a specific region of cortex at any moment (**Figures 6** and **7**) [44–46].

Intracranial recordings in experimental animals and humans for brain surgery (including epilepsy surgery) suggest that gamma (30–70 Hz) rhythms are mostly generated in the superficial somatosensory cortical layers II/III, while high beta rhythms (20–30 Hz) are mostly generated in deep layer V mainly in association with activity in the fast-spiking interneurons (**Figure 6**) [47]. The majority of cortical pyramidal cells are firing at a lower frequency range (usually <25 Hz) and their information coding may be based more on instantaneous firing rates or the critical timing and phase of their activation (timing or phase modulation code). Certain neurons in primary sensory (e.g. in early visual cortical areas) [48] and other neocortical or allocortical areas (e.g. in the hippocampus) can fire at higher frequencies and

#### **Figure 7.**

*Thalamocortical relationships. (A) The relative positions of thalamic nuclei. (B) Lateral (left) and medial (right) views of the cerebral cortex that demonstrate the projection targets of thalamic nuclei. Color-coding is to facilitate visualization of the reciprocal connections between thalamic nuclei and the cerebral cortex. VPM = ventral posteromedial nucleus; VPL = ventral posterolateral nucleus; VA = ventral anterior nucleus; VL = ventrolateral nucleus. Reproduced from http://what-when-how.com/neuroscience/the-thalamus-and-cerebral-cortex-integra tive-systems-part-2/.*

their information coding may be also based on the average firing rates (frequency modulation code) [46–49].

Gamma oscillations probably reflect a general rapid mechanism that 'synchronizes' neuronal networks in spatially disparate cortical areas to enable fast processing or coherent binding of visual, sensory or perceptual information across cortical and subcortical networks [50–52]. The integration of EPSP and IPSP into firing an action potential in a single neuron is essentially a nonlinear amplitudemodulation postsynaptic process and the majority of pyramidal neurons rely on an instantaneous or burst firing, critical-timing and phase-modulation code for information processing. Looking at neuronal networks from a macroscopic perspective (where intracranial LFP/EEG represents a collective mean-field measure), increased amplitude in higher-frequency oscillations may be considered as an index of neuronal spiking synchrony. That is, on a large scale there may be a quasi-linear relationship between the mean frequencies of oscillatory activities from postsynaptic integration of EPSPs and IPSPs on somatodendritic processes and the coordinated or synchronized firing rates of pyramidal cells, even if on the neuronal-cell microscale a non-linear relationship is seen through random or regular, bursting or tonic firing patterns of pyramidal cells [23, 25, 28, 29, 32, 37].

Although gamma oscillatory activities are also present during sleep and under anesthesia, fast oscillations are mainly associated with increased levels of alertness/ wakefulness, in keeping with activity in cholinergic neurons of the brainstem and basal forebrain [50, 53]. Increase of beta activity has been demonstrated after finger or foot movement when the muscles relax, while increase of gamma activity (greater than 30 Hz) immediately preceding the finger movement is thought to be associated with activation of cortical motor neurons. Sometimes on the scalp EEG we record equivalent sensorimotor cortical mu-rhythms which undergo suppression (desynchronization) upon performing a motor action or motor imagery, even observing another person performing a motor action or abstract motion [54].

Light sleep transition is characterized by alpha drop-out and appearance of Vsharp waves, spindles (alpha and beta range) and K-complexes (represent a depolarizing–hyperpolarizing sequence within an oscillatory cycle) before further transition to the less organised large slow wave oscillations of sleep (down to 1 Hz). Sleep spindles must be generated within the thalamus as they persist in the thalamus after decortication and high brainstem transection. Sleep spindles must be driven by the reticular nucleus of the thalamus (GABAergic neurons), as they are abolished in the dorsal thalamus after disconnection from the reticular nucleus but are preserved in the rostral part of the reticular nucleus severed from the dorsal thalamus [55].

At the onset of sleep decreased activity of brainstem cholinergic neurons contributes to the overall hyperpolarization of thalamocortical cells, thus bringing their membrane potential in the range where bursting discharges can occur. Such clusters of highfrequency action potentials excite the dendrites of neurons in the reticular nucleus, and trigger a dendrodendritic avalanche leading to synchronization of the entire reticular nucleus. Bursting of reticular neurons causes powerful GABAergic inhibitory postsynaptic potentials in thalamocortical neurons which promote cortical deafferentation (slow wave oscillations). The end of this inhibitory cycle a rebound low threshold spike (LTS) is triggered, crowned by a high-frequency burst of action potentials, which in turn excites the target reticular cells (**Figure 6**) [56–58]. Although previously thought that the main functional correlate of sleep spindles was to block incoming sensory stimuli from the thalamus to the cortex, today we believe that sleep spindles also serve a process of memory consolidation during sleep [59].
