Abstract

Physiological rhythms play a critical role in the functional development of living beings. Many biological functions are executed with an interaction of rhythms produced by internal characteristics of scores of cells. While synchronized oscillations may be associated with normal brain functions, anomalies in these oscillations may cause or relate the emergence of some neurological or neuropsychological pathologies. This study was designed to investigate the effects of topological structure and synaptic conductivity noise on the spatial synchronization and temporal rhythmicity of the waves generated by cells in the network. Because of holding the ability of clustering and randomizing with change of parameters, small-world (SW) network topology was chosen. The oscillatory activity of network was tried out by manipulating an insulated SW, cortical network model whose morphology is very close to real world. According to the obtained results, it was observed that at the optimal probabilistic rates of conductivity noise and rewiring of SW, powerful synchronized oscillatory small waves are generated in relation to the internal dynamics of cells, which are in line with the network’s input. These two parameters were observed to be quite effective on the excitation-inhibition balance of the network. Accordingly, it may be suggested that the topological dynamics of SW and noisy synaptic conductivity may be associated with the normal and abnormal development of neurobiological structure.

1  Introduction

The physiological and electrical rhythmic activities occurring in living beings are important phenomena associated with the vital biological functions. These biological rhythms are interacting with each other and also are affected by external sources. The feedback control caused by this mutual interaction and the environmental factors allow vital functions to be performed normally. It should be noted that healthy or pathological physiological functions are not directly associated with regular or irregular brain dynamics (Glass, 2001). Malfunction in vital mechanisms may cause some disorders, and some abnormal rhythms emerge from their normal limits (Niedermeyer & da Silva, 2005).

Many nonlinearly interacting complex and rapid spiky activities occur while the brain is functioning. It is known that these kinds of brain activities depend not only on the processes going on by sensory stimuli and internal mechanisms but also on the electrophysiological properties of the neurons and their synaptic interactions. The activity created, along with the external and internal inputs that modulate this activity, represent all vital phenomena and produce detailed response strategies (Destexhe & Contreras, 2006). It has been stated that slow-frequency rhythmic oscillations involve large neural regions, while high-frequency rhythmic oscillations are more localized (Penttonen & Buzsáki, 2003). In addition, previous studies show that not only geometric distance but also anatomical structure play effective roles in the spectra of brain waves (Csicsvari, Jamieson, Wise, & Buzsáki, 2003).

The functions of the large-scale synchronization of different brain regions are studied through single-cell recordings and local field potential (LFP) measurements in animals, and through more general measurements (such as EEG and EMG) in humans (Varela, Lachaux, Rodriguez, & Martinerie, 2001). It is suggested that some oscillations are occurring during wakefulness support cognitive functions, while some oscillations are associated with sleep mechanisms and long-term memory (Steriade, 1999). Neural synchronization as well as coherent oscillations are thought to regulate the coordination of and communication between neuron populations in simultaneously engaged cognitive processes (Wang, 2010).

Both in vivo and in vitro studies have shown that spindle oscillations persist in the thalamus without the involvement of the cortex. However, the neocortex plays important roles in propagating synchronized spindle activity to large areas in the cortex and in terminating separate spindle oscillations (Steriade, 2001). Contreras, Destexhe, Sejnowski, and Steriade (1996, 1997), from the experiments they conducted using intact cortex and isolated thalamus during barbiturate anesthesia and normal sleep in cats and humans, showed that spindles persist in the thalamus after decortications and the synchronization is disturbed over the cortex. However, in vivo measurements in cortical neurons showed that synaptic inputs have fairly vibrating characteristics (Destexhe & Sejnowski, 2003) and that spike discharges in a single cell do not have a fixed period and show a Poisson-like irregular distribution (Bair, Koch, Newsome, & Britten, 1994; Shadlen & Newsome, 1994; Softky & Koch, 1993). Measurements conducted on single cortical neurons of wakeful animals showed noisy spike activity resulting from fluctuations in membrane potential and irregular discharges within the 1–20 Hz range (Destexhe, 2009; Steriade, Timofeev, & Grenier, 2001). It is thought that this noise-dependent irregular spike form maximizes the information content in the neural brain code (Averbeck, Latham, & Pouget, 2006). It is also thought that it plays an important role in the stochastic behavior of biological mechanisms such as decision making and remembering, short-term memory, and attention (Deco, Rolls, & Romo, 2009) and pathological conditions such as schizophrenia and obsessive-compulsive disorder; it is also associated with the stability of the whole brain or a group of neurons within the population (Deco et al., 2009).

The synaptic background activity and noise significantly affect the electrophysiological properties and dendritic integration of neurons in cortical networks, which consist of a large number of neurons with dense synaptic connections (DeFelipe & Fariñas, 1992; Rudolph & Destexhe, 2001). Rudolph and Destexhe (2001), in their model study based on in vivo studies performed on cat parietal cortex (Destexhe & Paré, 1999; Paré, Shink, Gaudreau, Destexhe, & Lang, 1998), showed the causality of synaptic background noise to a stochastic resonance in pyramidal neocortical neurons. This study showed that the subthreshold periodical impulse intensity and correlations in background activity increase the responsiveness of the cells, similar to stochastic resonance, and that this may be associated with data processing in the cerebral cortex (Rudolph & Destexhe, 2001).

Deterministic noisy signals may be quantified with some statistical measures such as stochastic resonance (SR) (Schmid, Goychuk, & Hänggi, 2001). Noisy nonlinear systems may also show coherent resonance or an SR-like behavior called autonomous SR (Wang, Chik, & Wang, 2000). Coherent resonance phenomena in globally connected Hodgkin-Huxley neurons were first investigated by Wang et al. (2000). In that study, they observed that no spatiotemporal order exists when the synaptic connection strength is low and the coherence increases at first if the synaptic connection strength (linearly correlated with the synaptic noise) is large enough, but it decreases after a certain level of synaptic noise. Stacey and Durand (2000), in their in vitro experimental and model studies, investigated the effects of noise in hippocampal CA1 cells and found that the detection of subthreshold weak synaptic inputs could be improved with induced noise. In another study (Stacey & Durand, 2002), they showed that physiological noise and electrical synaptic connections in hippocampal CA1 neurons improve signal detection.

From a computational biology point of view, the first topological studies concerning the mammalian cortex were carried out at the beginning of the 1990s. The findings from these studies conducted on monkey (Felleman & Van Essen, 1991; Young, 1992, 1993) and cat (Young, 1992, 1993) visual cortices show many strong connections between neighboring regions and fewer connections between distant brain regions. This form of anatomical structure is consistent with the SW model (Bassett & Bullmore, 2006) developed by Watts and Strogatz (1998) many years later. The study by Hilgetag and Grant (2000) states that the large-scale monkey and cat cortices show SW properties with their high level of clustering and low path length characteristics. Analysis performed on the results obtained from magneto-encephalographs (MEG) (Stam, 2004), electroencephalographs (EEG) (Micheloyannis et al., 2006; Stam, Jones, Nolte, Breakspear, & Scheltens, 2007), and magnetic resonance images (fMRI/MRI) (Bassett et al., 2008; He, Chen, & Evans, 2007) of the human brain show that the connection pattern between functional areas of the brain has an SW structure with shortcut connections between different regions and clustered local connections (Bassett & Bullmore, 2006). These experimental studies also show that the brain develops to achieve complex information processing with the lowest connections cost.

All in all, understanding the physiological mechanisms for self-emerging oscillations in brain may help in both understanding the functions of these oscillations and diagnosing and treating brain disorders. In this regard, considering that both the SW rewiring probability and the noisy synaptic conductivity play an important role in the emergence of coherent and synchronized spike discharges, an important question to answer is how these parameters affect slow, synchronized rhythmic oscillations. It is almost impossible to rearrange the synaptic connections or modify the noisy synaptic conductivity among a specific cell population experimentally. At this point, computational models are very convenient tools.

This study investigates the effects of these two parameters on the developed SW network output. The SW model was developed for the isolated cortical system solely to analyze the behavior of the cortical network. In the study, the dynamics of slow oscillations at the output of SW, which in reality emerge especially during deep sleep, were analyzed with respect to the rewiring probability ( and standard deviation of noisy synaptic conductivity (. As a result of these analyses, it is observed that the sensitivity of cortical cell groups to sensory inputs and internal dynamics increases, and at some SW and values, the rhythmic oscillatory activities of the network become more synchronized as well as coherent. These results are consistent with the results obtained from both experimental and computational methods in the literature.

2  Materials and Methods

Based on in vitro and in vivo experimental studies, cell models and, accordingly, generalized neocortex network models with some basic parameters have been developed (Mainen & Sejnowski, 1996). The neocortex network model consists of pyramidal (PY) and interneuron (IN) cells. The cortical cells are represented with voltage- and calcium-dependent channel currents holding two-compartment Hodgkin--Huxley-like kinetics (Hodgkin & Huxley, 1952) scaled by absolute temperature (Coulter, Huguenard, & Prince, 1989).

2.1  Computational Models of Intrinsic Currents

Cortical PY and IN cells have two compartments (axosomatic and dendritic) connected through axial resistance (. The firing pattern of PY and IN cells depends on the parameter that represents the ratio of the dendritic compartment to the axosomatic compartment. The burst discharge in PY cells and the regular spiking (RS) discharge pattern in IN cells have the values , and , respectively (Timofeev, Grenier, Bazhenov, Sejnowski, & Steriade, 2000). Mainen and Sejnowski’s (1996) study was used as the base for both cell models. The cell models employ a simplified version (Pinsky & Rinzel, 1994) of a two-compartment complex detailed cell model described for hippocampal CA3 pyramidal cells by Traub, Wong, Miles, and Michelson (1991). To increase burst activity in PY cells (Timofeev et al., 2000), this model adds persistent sodium current( in both compartments (Alzheimer, Schwindt, & Crill, 1993). The two-compartment general cell model described for PY and IN cells is defined by equation 2.1 (Bazhenov, Timofeev, Steriade, & Sejnowski, 1998b, 2002; Timofeev et al., 2000),
formula
2.1
where is the neuron index; and are the membrane capacitance of dendritic and somatic compartments, respectively ( for each cell type and each compartment); is the leak current conductivity ( for both cortical cell types; membrane resistance); is the leak current reversal potential ( for PY and IN cells); and and are the membrane potentials for dendritic and axosomatic cells. and are the external DC or other external input in different forms, applied to the dendritic and axosomatic compartments, respectively; and are the internal currents that the dendritic-and axosomatic compartments; and is the synaptic current (defined between neuron and neuron of dendritic compartments based on synaptic conductivity and membrane potential. synaptic conductivity is calculated by adding noise to the point-conductivity model based on equation 2.13, similar to the Ornstein-Uhlenbeck stochastic process. and are dendritic and somatic axial conductivities between compartments. The length of axosomatic compartments of PY and IN cells is , their diameter is , while their surface area is ; the length of the dendritic compartments is , and its width is ; their surface area is (Bazhenov et al., 1998b, 2002; Mainen & Sejnowski, 1996; Timofeev et al., 2000). The dependent parameters for both PY and IN cells must be calculated again, based on the new value. The required parameter values for the calcium dynamics for both cells are taken as time constant , thickness of the shell below the membrane , extracellular calcium concentration and extracellular steady-state calcium concentration (Bazhenov et al., 1998b, 2002).

2.1.1  Pyramidal Cell Model

The pyramidal cell model has dense channels in the axosomatic compartment, and low-density channels (fast current) in the dendritic compartment. Both compartments have persistent sodium current (, slow voltage-dependent noninactivating current allowing frequency adaptation (, slow -dependent current (, high-threshold current ( or , and fast delayed rectifier current (. Additionally, the dendritic compartment has leak current and low-threshold current ( (Bazhenov et al., 1998b, 2002; Timofeev et al., 2000):
formula
2.2
Here, the voltage- and calcium-dependent currents and kinetic definitions that have been described by Mainen and Sejnowski (1996), Bazhenov et al. (1998b, 2002), and Timofeev et al. (2000) were used (for both compartments, , , , . The maximum conductivity and reversal potential values for axosomatic and dendritic compartments are presented in Table 1.
Table 1:
PY Cell Maximum Conductivity and Reversal Potential Values.
Axosomatic compartmentDendritic Compartment
Maximum Conductivity (Reversal Potential (Maximum Conductivity (Reversal Potential (mV)
    
    
    
    
    
    
    
    
Axosomatic compartmentDendritic Compartment
Maximum Conductivity (Reversal Potential (Maximum Conductivity (Reversal Potential (mV)
    
    
    
    
    
    
    
    

Sources: Bazhenov et al. (1998b, 2002); Timofeev et al. (2000).

As PY cells have AMPA and NMDA synaptic connections among themselves and GABA synaptic connections from IN cells, the total of synaptic currents for each cell is represented as (Bazhenov et al., 1998b, 2002)
formula
2.3
, and are represented between PY-PY , PY-PY , and IN-PY synaptic currents, respectively. , , and maximum receptor conductivity values are given in Table 3, and their reversal potentials are , , and , respectively (Bazhenov et al., 1998b, 2002).

2.1.2  Interneuron Cell Model

The interneuron cell (IN) model, like the PY model, has high-density channels in the axosomatic compartment, low-density channels (fast current) in the dendritic compartment, and a fast delayed rectifier current ( only in the axosomatic compartment. Moreover, the dendritic compartment has leak current , slow voltage-dependent noninactivating current allowing frequency adaptation (, slow -dependent current (, and high-threshold current ( (Bazhenov et al., 1998b, 2002; Timofeev et al., 2000).
formula
2.4
Here, as in the PY model, the voltage- and calcium-dependent currents and kinetic definitions that have been described by Mainen and Sejnowski (1996), Bazhenov et al. (1998b, 2002), and Timofeev et al. (2000) have been used for the IN cell model (for both compartments, , , , . The maximum conductivity and reversal potential values for axosomatic and dendritic compartments are presented in Table 2.
Table 2:
IN Cell Maximum Conductivity and Reversal Potential Values.
Axosomatic CompartmentDendritic Compartment
Maximum conductivity (Reversal Potential (mV)Maximum Conductivity (Reversal Potential (mV)
    
    
    
    
    
Axosomatic CompartmentDendritic Compartment
Maximum conductivity (Reversal Potential (mV)Maximum Conductivity (Reversal Potential (mV)
    
    
    
    
    

Sources: Bazhenov et al. (1998b, 2002); Timofeev et al. (2000).

As IN cells have AMPA and NMDA synaptic connections from PY cells, the total of synaptic currents for each cell is represented as (Bazhenov et al., 1998b, 2002)
formula
2.5
and represent between PY-IN and PY-IN synaptic currents, respectively. and maximum receptor conductivity values are given in Table 3, and their reversal potentials are and , respectively (Bazhenov et al., 1998b, 2002).
Table 3:
Maximal Synaptic Conductivity Values between Neurons.
Receptor TypeDirectionOptimal Conductivity Value
   
   
   
   
   
Receptor TypeDirectionOptimal Conductivity Value
   
   
   
   
   

Sources: Bazhenov et al. (1998a, 2002).

2.1.3  Calcium Dynamics

Calcium dynamics for calcium-dependent currents are represented with a first-order differential equation that describes the pump and its buffer (Destexhe, Contreras, Sejnowski, & Steriade, 1994),
formula
2.6
where is the time constant, is the steady-state intracellular concentration, is the Faraday constant, is the thickness of the shell below the membrane, and is the total of low-threshold ( and high-threshold (-dependent currents in the cell model (Halnes, Augustinaite, Heggelund, Einevoll, & Migliore, 2011). As the intracellular concentration varies according to the calcium currents, the calcium reversal potential needs to be recalculated at every step, based on the Nernst equation (Hille, 2001).

2.2  Synaptic Currents

The kinetic models for the synaptic currents used in this study are based on experimental data recorded from hippocampal neurons (Otis, De Koninck, & Mody, 1993; Otis & Mody, 1992), and models described in a series of studies by Destexhe, Mainen, and Sejnowski (1994a, 1994b, 1998) and Destexhe, Bal, McCormick, and Sejnowski (1996).

The exhibiting and receptors and the inhibiting receptors are represented simply according to the first-order activation scheme as (Destexhe et al., 1996)
formula
2.7
where is the number of transmitters released in the synaptic cleft or bound to receptors. The ratio constants are voltage independent and have the values , for , , for (Destexhe, Mainen et al., 1998). Transitions between the open and closed states depend on the amount of receptor-bound transmitters and are represented, depending on the presynaptic voltage (, as (Destexhe, Mainen et al., 1998)
formula
2.8
where is the maximum transmitter concentration (, is the presynaptic voltage, is the value at which the function is half-activated, and is the variable ensuring steepness. concentration may be thought of as an impulse depending on the duration of the presence of neurotransmitters in the synaptic cleft, following the AP in the presynaptic neuron (approximately and with an approximate amplitude of (Destexhe et al., 1996). The ratio of receptors in the open state being , the corresponding model is represented as (Bazhenov, Timofeev, Steriade, & Sejnowski, 1998a; Destexhe et al., 1996; Destexhe, Mainen et al., 1998)
formula
2.9
and, finally, the postsynaptic and currents are represented as (Destexhe, 1998; Destexhe et al., 1994a)
formula
2.10
where , , , , and represent the postsynaptic membrane potential, ratio of receptors in the open state, maximum conductivity, the transmitter concentration released in the synaptic cleft, and the reversal potential, respectively. and represent forward and backward binding rate constants of the transmitters activating the receptors, respectively (Destexhe, 1998; Destexhe, Mainen et al., 1998).
receptor-dependent synaptic currents need two agonist molecules to open the receptors because they are slower than currents current is modeled based on the two-state model similar to receptor model AMPA and described by adding the voltage-dependent term to represent magnesium blockage (Destexhe, Mainen et al., 1998; Murray et al., 2014)
formula
2.11
and
formula
2.12
where is the maximum conductivity and is the reversal potential. represents voltage-dependent magnesium blockage and is the extracellular magnesium concentration, with a value between 1 and 2 mM. In line with experimentally proven data for current, they have the values , , and (Destexhe, Mainen et al., 1998).

2.3  Synaptic Conductivity Noise

The synaptic background activity, which may be called noise in intracellular records of in vivo neurons, is always present (Destexhe, Rudolph, Fellous, & Sejnowski, 2001). In this study, a noise term has been added to the synaptic conductivities ( similar to the Ornstein-Uhlenbeck stochastic process (Uhlenbeck & Ornstein, 1930) according to the point-conductivity model, and this term is represented for each synaptic receptor type as (Destexhe et al., 2001),
formula
2.13
where is the maximum synaptic conductivity amplitude coefficient (different for each synaptic connection type) and is the white gaussian noise with zero mean and unit standard deviation. and are the diffusion coefficients of the process and the time constant. The variance of the stochastic conductivity ( obtained from equation 2.13 is (Destexhe et al., 2001). In this study, the noise diffusion coefficient ( is calculated as dependent on the noise conductivity standard deviation ( that changes by steps of 0.01 within the range , with the time constant being .

2.4  Network Geometry

Anatomical studies show that the axonal projections of cortical cells may be local and topographical, and each cell type may make synaptic connections with other cell types in a ratio based on the network population (Destexhe, Contreras, & Steriade, 1998). To be able to observe the effects of the SW model and the synaptic conductivity noise on the spatiotemporal dynamics of the network in the cortical transformation, the synaptic connection effective area (radius) in both directions has been limited to a certain ratio of the population of the projecting cell type (Chen, Chauvette, Skorheim, Timofeev, & Bazhenov, 2012). Compared to a real network, only size-limited modeling can be performed. The network simulation depends heavily on synaptic conductivities, and changes in synaptic conductivity affect the network’s behavior (Destexhe, Contreras et al., 1998). Selecting the appropriate values for the synaptic conductivities (or weights) is one of the greatest difficulties. Thus, in this study, the network sizes and synaptic weights have been determined based on the values previously used in the studies by Destexhe et al. (1996), Destexhe, Contreras et al. (1998), and Bazhenov et al. (1998a, 1998b, 2002) concerning rhythmic oscillation behavior. The values in Table 3 represent the maximal conductance that may be formed from a receptor type to any cell.

One hundred pyramidal ( and 25 inhibitor ( neurons have been used by adhering to the 4:1 ratio (Bazhenov et al., 2002) for the number of PY and IN cells in the cortical network for compliance with anatomical and morphological fundamentals. For all synaptic connection types, the effective connection area in the direction of the projection target cell (, radius ratio) is limited to 5% of the cell type that is projecting the connection (Bazhenov et al., 2002). The synaptic projections between any given cell type, synaptic connection type, and conductivities are chosen to be the same throughout the network.

2.5  Data Processing

Various methods were used to analyze the data obtained at the end of the simulation in both the spectral and temporal domains in this study. To represent the total electrical activity of cell groups instead of a simple and widely used LFP model that considers the current sources in the homogeneous extracellular environment (Nunez & Srinivasan, 2006), a more realistic LFP model proposed by Bédard, Kröger, and Destexhe (2004), which considers frequency-dependent weakening, was used. The methods used include population spike firing synchronicity based on the spike emergence times as a measure of synchronization and irregularity, vector Strength, and spike frequency. The spectral characteristics of the total electrical activity (LFP) of cell populations in the network in the frequency domain were investigated, as was the frequency range in which the rhythmic oscillations resonate using stochastic resonance and coherence resonance methods.

2.6  Stochastic Resonance

To calculate SR value, the spectral power distributions (PSD) of the cumulative indicators obtained from the postsynaptic potential (PSP) created by synaptic currents of the cells (generally pyramidal) according to the LFP model have been used. The voltage response of the network (LFP) is obtained by detrending the rest of the membrane potential values of each neuron to zero. The power spectrum distribution has been obtained by Welch’s averaged periodogram method, using the Hanning window ( overlap), which has a sampling frequency , and length (frequency resolution, (Welch, 1967). While simultaneous spikes cause high-amplitude outputs, spikes that occur in different times and do not overlap cause outputs that are spread over a wide temporal region (Stacey, Lazarewicz, & Litt, 2009). The SR has a relation with the noise and, hence, the signal-to-noise ratio (SNR) as in equation 2.14 given by Stacey and Durand (2000). From the equation, it can be seen that from a certain minimum level (or zero), SNR increases with the increase of added noise and reaches its peak at an optimum noise level and then gradually falls back to one (Stacey & Durand, 2000, 2002),
formula
2.14
where is the noise intensity, is the signal strength, and is the threshold barrier height. The dependence of SR on noise level has a characteristic similar to the SNR defined in this way, and therefore the SR value is commonly represented by such a value calculated by dividing the spectral power value at the input (periodical) frequency to the fundamental power value representing the spectral noise around this frequency value (Stacey & Durand, 2002; Stacey et al., 2009).

2.7  Vector Strength

Vector strength (VS) is represented in the neuronal system as a synchronized phase-locked spike activity (Ashida & Carr, 2010). The VS is the measure of the phase locking or synchronization of a series of spikes (Grün & Rotter, 2010). It is a measure of phase locking within the range , based on a reference frequency ( (or the approximate frequency of the system). In this method, each spike is considered as a unit vector with phase angle ( is the emergence time of the th spike) (Ashida & Carr, 2010). VS takes a value between zero and one for a certain reference period; values near one mean strong phase locking (synchronization), and values near zero mean as irregular spike series (Ashida & Carr, 2010).

2.8  Population Spike Coherence

The method proposed by Wang and Buzsáki (1996), which considers the normalized cross-correlations (CC) of the neurons in the network, has been used as a measure of coherence to determine the simultaneous spike activity measure of the neurons on the network. The coherence between any two neurons in the network ( and is measured by the normalized cross-correlation between the spike firing times determined based on zero time delay and a certain time window ( (Wang & Buzsáki, 1996). The population coherence measure ( has a value between zero and one, and it tends toward one as the network synchronization increases.

2.9  Application and Computational Methods

The focus of this study is to investigate the effects of rewiring probability ( and the standard deviation of noisy synaptic conductivity ( on electrical oscillations at the output of the SW model developed for representing an isolated cortical model. The analyses were conducted based on the assumption that the cortex is isolated from other brain regions. The cell connection topologies were first designed as regular ring shapes (, and then the existing connections were restructured based on the SW rewiring probability , which changed from zero to one in 0.1 increments. Here, 0.1 represents of the existing connections, and 1 represents . In this way, by varying the connection topologies of the SW network, its characteristic changes from a regular network to a completely irregular one. To analyze the effects of synaptic conductivity noise over the network, the noisy conductivity standard deviation ( value was increased in steps of 0.01 between 0 and 0.1. Thus, the noisy synaptic conductivity ( for each synapse type was calculated according to the point-conductivity model given by equation 2.13, based on the stochastic component diffusion coefficient ( and the maximum synaptic conductivity amplitude coefficient (. Periodic (0.5 Hz) -type external synaptic inputs having normal (gaussian) spike distribution activity in each period were applied to the cortical cells to examine the behavior of rhythmic electrical oscillations and the coherence of spikes generated by cells with respect to the change in synaptic conductivity noise and the change in topology of the isolated cortical SW network model. There were 121 possible combinations, since both of variables are considered. The effects of 121 different combinations of these two parameters were investigated through simulations, each of which took 8 seconds (with 0.1 ms steps).

The initial voltages, intracellular calcium concentrations, and calcium reversal potentials of the cortical cells were determined randomly within appropriate physiological ranges given in Table 4 to obtain stable results. The power function distribution (Kleiber & Kotz, 2003) was used as a basis for the random process, and the selected value ranges, and were set to fall within the range [0–1], which is the power function distribution range. Moreover, to ensure accurate results for randomly determined parameters within a fixed or specified range, all simulations were taken as the same for all and parameter values, and each trial was run with the same initial conditions.

Table 4:
Initial Parameter Values for Cell Groups in Cortical Network Models.
CellParameterValue Range
PY   
   
   
IN   
   
   
   
CellParameterValue Range
PY   
   
   
IN   
   
   
   

The implementation was made on an HP Z820 workstation with 2 Xeon E5-2620 (2 GHz 12 Core) processors, 2 8GB DDR3-1600 RAM + 40 GB Linux swap and 2 450GB SAS 15K hard disks running the Ubuntu 12.04 (64-bit) server operating system, using the Python programming language and the Brian library (Goodman & Brette, 2008, 2009).

3  Results

The regular (lattice) network model in Figure 1 was designed initially to model the rhythmic burst activity in the cortical network. The number of PY cells was , and the number of IN cells was in the network. To approximate external synaptic inputs to thalamic inputs, it was assumed that the inputs came from the same source for both cell types. In Figure 1, the number of synaptic inputs for each neuron is the radius ratio of the total number of neurons in the neuron type (PY or IN) that is projecting the synaptic connection toward itself in the direction of the arrow (Destexhe, Contreras et al., 1998).

Figure 1:

Regular ring-shaped connection structure between cortical cell groups. Each PY neuron has AMPA and NMDA synaptic projections from neighboring PY neurons, and GABA synaptic projections from IN neurons, whereas IN neurons only receive AMPA and NMDA synaptic projections from PY neurons . The values for synaptic conductivities are , , , , and (Bazhenov et al., 2002).

Figure 1:

Regular ring-shaped connection structure between cortical cell groups. Each PY neuron has AMPA and NMDA synaptic projections from neighboring PY neurons, and GABA synaptic projections from IN neurons, whereas IN neurons only receive AMPA and NMDA synaptic projections from PY neurons . The values for synaptic conductivities are , , , , and (Bazhenov et al., 2002).

To obtain an oscillating rhythmic activity, a certain number ( of unrelated artificial periodic (0.5 Hz) spike activities with the maximum synaptic conductivity strength were applied to the isolated cortical network. The spontaneous firing activities of the spikes applied in each period were in a certain frequency (5 Hz) and had normal (gaussian) distribution (. Each cortical PY and IN neuron and the artificial spike series were associated with the input at a certain ratio ( of all external inputs) using AMPA synaptic receptors. The cortical network model was run for 8 seconds (8000 ms), and each cortical network group (PY and IN) was analyzed on its own, considering the activity after the first second (1000 ms).

3.1  Changes in Activity

The excitatory source was chosen to be the same for all sensory inputs of the cortical network, as if the network were excited by thalamic neuronal cells. In this way, the effects of changes in the parameters and on the activities and input sensitivities of cortical PY and IN cells were investigated. Figure 2 shows the spike and passive cell counts depending on the parameters and , and it can be seen that almost all the cortical PY cells are passive up to a certain rewiring probability value (. Although all cells are active in the IN cell group, it is evident that the spike activity is similarly weak within this range. As observed, the responsiveness and, consequently, the network activities of the cortical cell groups increased dramatically as increased in the cortical model. The -related activity increase resulted in associated synaptic activity and consequently increased the influence of on network activity. At suitable and values, both PY and IN cell activities reached their highest-frequency values and exhibited maximum activity.

Figure 2:

Cortical network active and passive cell counts and spike frequency variations. The changes in the total spike counts, passive cell counts, and average frequency values of 100 PY and 25 IN cells in the cortical cell model as a result of 8 seconds of simulation (except the first second) in relation to the parameters and .

Figure 2:

Cortical network active and passive cell counts and spike frequency variations. The changes in the total spike counts, passive cell counts, and average frequency values of 100 PY and 25 IN cells in the cortical cell model as a result of 8 seconds of simulation (except the first second) in relation to the parameters and .

Figure 3 shows the changes in the spike activity of the cortical network in relation to some and values. In Figure 3a, the total spike count and average frequency for the cortical PY cell group for all values is seen starting from a value near zero and increasing nonlinearly as the probability value increases. The activity reaches a peak value for some values. For probability values of , many PY cells (up to 80%) have passive or very weak activity; as the probability increases, the passive cortical PY cell count drops sharply in line with the network activity and becomes almost zero after this probability value (. This shows that at least 20% of the existing synaptic connections must be rewired according to the SW model in order to increase the input sensitivity of the PY neurons in the network and ensure homogeneity in terms of spike activity. While cortical IN cells exhibit a certain level of activity for all and values, the activity of these cells also increases with the parameter, and the activity curves are parallel to the PY cells. All activity curves that visibly diverge in Figure 3b are parallel to the horizontal axis and exhibit almost linear change, while there are sharp peaks for different values of the curves belonging to different values. The activity—in other words, the input sensitivity—reaches the maximum values at suitable (optimum) and values. Additionally, there are critical and values that relatively increase the passive cell count and decrease activity. The input sensitivity or responsiveness of the cortical network showed more significant increases for suitable and values, although the input activity was the same. Accordingly, when the wiring between neurons is randomly severed and randomly rewired, it is observed that the noisy synaptic conductivity activity that varies with respect to synaptic connections synchronizes the subthreshold asynchronous spike activities, thus increasing input sensitivity.

Figure 3:

Cortical network active and passive cell counts and spike frequency variations for different and values. Activity change graphs for the network in Figure 2(a) in relation (a) p for the and 0.06 values and (b) for the and 0.9 values.

Figure 3:

Cortical network active and passive cell counts and spike frequency variations for different and values. Activity change graphs for the network in Figure 2(a) in relation (a) p for the and 0.06 values and (b) for the and 0.9 values.

The voltage traces (for a single cell) and the spatial and temporal spike activity changes for some and value pairs for which the activity of cortical PY and IN cell groups relatively increases or decreases are presented in Figure 4. Figure 4a shows that the cortical network activity is strengthened and AP discharges increase as the probability increases, and at suitable values, although the periodical AMPA synaptic activity is constant. Figure 4b shows temporal and spatial frequency changes. Table 5 shows that while there are weak, subthreshold fluctuations in the value pair, the cortical PY and IN cell spike activities are weak because the temporal and spatial average frequency values are very low. In contrast, the significant increase in the temporal and spatial frequency values for and value pairs of cortical PY and IN cells shows that they have higher spike activities. Here, it can be observed that there is no linear relationship between the activity and standard deviation of noisy synaptic conductivity at the same SW probability (, while the activity is strengthened for some optimal values.

Figure 4:

Voltage traces and temporal and spatial spike activities for various and value pairs in the cortical network. (a) Voltage traces and (b) temporal and spatial frequency changes for , , and value pairs of the cortical network model (for 7 seconds, excluding the first second). The temporal and spatial frequencies represent the spike frequencies during simulation in relation to time and neuron, respectively. Subthreshold activities are represented with an arrow in the voltage traces. The red line represents the average value in the spatial frequency curves and distribution graphs. , , , and denote the mean and standard deviation of the temporal and spatial frequencies, respectively.

Figure 4:

Voltage traces and temporal and spatial spike activities for various and value pairs in the cortical network. (a) Voltage traces and (b) temporal and spatial frequency changes for , , and value pairs of the cortical network model (for 7 seconds, excluding the first second). The temporal and spatial frequencies represent the spike frequencies during simulation in relation to time and neuron, respectively. Subthreshold activities are represented with an arrow in the voltage traces. The red line represents the average value in the spatial frequency curves and distribution graphs. , , , and denote the mean and standard deviation of the temporal and spatial frequencies, respectively.

Table 5:
Average Activity and Standard Deviation Values of Cortical Cell Groups for Various and Value Pairs.
ValuesTemporal Frequency (TF)Spatial Frequency (SF)
Cell       
PY 0.1 0.03 1.61 4.4 2.17 2.03 
 0.7 0.03 5.25 9.89 5.96 4.66 
 0.7 0.01 7.05 13.1 7.83 6.49 
IN 0.1 0.03 10.19 14.97 10.86 1.76 
 0.7 0.03 18.86 24.33 19.81 3.61 
 0.7 0.01 19.7 26.11 20.73 4.59 
ValuesTemporal Frequency (TF)Spatial Frequency (SF)
Cell       
PY 0.1 0.03 1.61 4.4 2.17 2.03 
 0.7 0.03 5.25 9.89 5.96 4.66 
 0.7 0.01 7.05 13.1 7.83 6.49 
IN 0.1 0.03 10.19 14.97 10.86 1.76 
 0.7 0.03 18.86 24.33 19.81 3.61 
 0.7 0.01 19.7 26.11 20.73 4.59 

The standard deviation values for the temporal ( and spatial ( frequencies given in Table 5 for the value pair of cortical PY and IN cells are relatively low due to low activity. Accordingly, it can be stated that the temporal activities of the cells are spread over the time domain, and both the temporal and the spatial frequency values of the cells are close to each other and the average values. Therefore, the activities of the cell groups are very low for this parameter pair, and, consequently, wave-shaped oscillations with active-silent state transitions are very weak, based on the evaluation of the temporal and spatial change curves (shown in Figure 4). In contrast, the standard deviation values for the temporal and spatial frequencies given in Table 5 for , and value pairs are high due to increased activity, and wave-shaped oscillating rhythmic activities that are consistent with periodic (0.5 Hz) input are formed as observed in the sample voltage traces and graphics in Figure 4. Although the relative irregularity in the temporal and spatial spike activities is expected for these value pairs where the activity increases dramatically, a strong oscillating activity dominated by the input activity emerged in cortical PY and IN cell groups. Moreover, while the PY spike activity increased for the value pair, the electrophysiological characteristics of the cells did not change, and they continued to generate rhythmic burst (intrinsic bursting, IB) spike discharges. In contrast, it is observed that the PY cells generate high-frequency spikes (fast spiking, FS, or fast rhythmic bursting, FRB) discharges for the value pair.

The figures in bold in Table 5 show that the highest averages and of TF and SF for both PY and IN cell groups come about at and pairs, while the lowest standard deviations ( and ) of TF and SF for the cell groups occur at and pairs.

3.2  Phase-Locking

The VS method is used to analyze the temporal and spatial synchronizations of successive spikes generated by cortical PY and IN cells in response to the applied periodic input. This method evaluates the spike activities of cortical PY and IN cells separately and all together, and it was used to analyze the intercellular coherence and the periodicity of successive spike series based on the external periodical input (0.5 Hz) that dominates the cortical network. Figure 5a shows VS and its distribution obtained for cortical PY and IN cell groups with the and value pairs used in previous analyses, where the red line shows the average value.

Figure 5:

VS analyses for various and value pairs in the cortical network. VS analyses of the cortical PY and IN cell groups for the and value pairs used in Figure 4 are presented. (a) VS curves and VS distribution graphs of cell groups (the red line indicates the average VS values). In the VS distribution graph, the -axis indicates the number of neurons. and denote the mean and standard deviation, respectively, of VS. (b) The spike activity of the whole network groups in the polar domain at the 1:100 scale, radial phase distribution, and phase pattern based on the reference period, which is the external AMPA input activity frequency (0.5 Hz).

Figure 5:

VS analyses for various and value pairs in the cortical network. VS analyses of the cortical PY and IN cell groups for the and value pairs used in Figure 4 are presented. (a) VS curves and VS distribution graphs of cell groups (the red line indicates the average VS values). In the VS distribution graph, the -axis indicates the number of neurons. and denote the mean and standard deviation, respectively, of VS. (b) The spike activity of the whole network groups in the polar domain at the 1:100 scale, radial phase distribution, and phase pattern based on the reference period, which is the external AMPA input activity frequency (0.5 Hz).

In the figure, for , the VS for each cortical PY and IN cell and the averages of these values are significantly low. The VS distributions are better clustered when is close to zero. As stated in previous analyses, this situation is caused by very low spike activity at this value pair. From the polar distribution obtained for the cortical PY and IN cell groups at presented in Figure 5b, it can be said that the spikes are clustered inside a narrow radial range, which shows that the spike discharges are better synchronized. For and value pairs, the activity of cortical PY and IN cells gradually increased, while spikes are clustered in a slightly wider range. However, despite the increased activity, VS values tended to move away from zero. Accordingly, at some and values, it is observed that both spatial synchronization and the temporal rhythmic or periodical spike activity are strengthened. Despite the same external periodic input, which has an irregular spike activity in each period, at appropriate and values, the spatial synchronization and the temporal periodicity increased along with the activity. It is thought that this is caused by the resonance due to the synchronous emergence of weak subthreshold PSP activities arising from the internal rhythmic characteristics of the cells and the consequent increase of their influence over the cell activity.

3.3  Population Spike Coherence

The simulation duration in the cortical network ( seconds, the duration excluding the first second) is divided to time intervals, and the spike coherence of the population ( is obtained accordingly. In Figure 6, the spike coherence measure ( of the cortical PY and IN cell groups increases proportionally with the increase in the parameter , while it has no definite association with the parameter , and it exhibits maximum values at optimal regions.

Figure 6:

Population coherence measure of the cortical network. (a) Spike coherence measure patterns in relation to the change in and parameters. (b) Coherence change curves in relation to for , and 0.06 values and in relation to for , and 0.9 values.

Figure 6:

Population coherence measure of the cortical network. (a) Spike coherence measure patterns in relation to the change in and parameters. (b) Coherence change curves in relation to for , and 0.06 values and in relation to for , and 0.9 values.

Figure 7 shows the spike and CC average maps for the value pairs used in previous analyses. According to the spike and CC average maps for the cortical PY and IN cells for the pair and the spike coherence measure ( in Table 7, the low-association density in the maps shows that both the spike activity and the correlation between the existing spike activities are weak. The association that is sparse (for PY) and clustered diagonally on the CC average map for this value pair shows that some of the cells in the cortical PY and IN groups are passive, and others have strong spike associations only with a few neighboring cells while being weakly associated with other cells. Additionally, according to the spike coherence measure values in Table 7, the correlation between both the activities and the spikes of cortical PY and IN cells has increased for suitable and values. Figure 7 shows that the sparsity due to passive cortical PY cells is decreased and that there are strong associations especially among IN cells, as evidenced by the spike and CC average maps of and value pairs. The increase in the spike correlations of cortical PY and IN groups shows that the spike coherence between these cells is spread over larger areas and is further strengthened.

Figure 7:

CC and spike maps for various and value pairs in cortical network. CC and spike maps for , , and value pairs. Each row and column of the CC map pattern corresponds to a neuron. Warm (cool) colors in the CC map pattern indicate a high (low) coherence measure based on the normalized cross-correlations of neuronal pairs in the network group (PY or IN). The spike map shows the presence of the spiking activity in a simulation time interval divided into small bins (.

Figure 7:

CC and spike maps for various and value pairs in cortical network. CC and spike maps for , , and value pairs. Each row and column of the CC map pattern corresponds to a neuron. Warm (cool) colors in the CC map pattern indicate a high (low) coherence measure based on the normalized cross-correlations of neuronal pairs in the network group (PY or IN). The spike map shows the presence of the spiking activity in a simulation time interval divided into small bins (.

3.4  Spectral Analyses

In order to analyze the output of SW cortical network for total LFP created by the PSP activities of cortical PY and IN cell groups, maximum spike frequency, PSD values, and stochastic resonance of the LFPs were used Figure 8 shows the rate of change of maximum frequency, PSD, and SR of LFPs with respect to changes of and values of cortical PY and IN cell groups in response to a periodic irregular input. For all and values, the frequency at maximum amplitude values of the PSD for LFPs of cortical PY and IN cell groups is near the frequency value of the periodic input 0.5 Hz and that the maximum frequency patterns are almost the same for these two groups. The subthreshold weak vibrations due to the input cause an oscillating activity in the LFP, albeit with low amplitude, at the low and values where the input sensitivity and spike activity are weak. In Figure 8, the maximum PSD values of the cortical PY and IN cell groups have increased in the same ratio with the activity increase due to the value. The SR values are high in the figure wherever the activity and maximum PSD values are high due to the strong resonance of cortical PY and IN cells. Also, the dependence of stochastic resonance in the small-world isolated cortical network on the rewiring probability for various levels of PSD of synaptic noise can be observed in Figure 8. The value of stochastic resonance keeps increasing as the rewiring probability becomes larger. It is shown that stochastic resonance saturates almost when ; that is, beyond this limit, can have little effect the peak of stochastic resonance. To better show this result, Figure 9 is provided. It can thus be confirmed that the rewiring probability has a significant impact on the stochastic resonance when . Increasing the rewiring probability can always enhance the stochastic resonance before it approaches the limit beyond which the network becomes random. The cells within the cortical PY and IN cell groups resonate at the input frequency at suitable (around 0.2) and value pairs. As seen from the average maximum SR and maximum PSD graphs in Figure 9, SR and PSD rates almost reach their steady state as for almost all values of ; however, while the PSD negligibly decays, the SR rate moderately increases for .

Figure 8:

Maximum frequency, PSD, and SR changes in the cortical network. Maximum frequency (Hz), PSD (dB/Hz), and SR change graphs calculated based on and parameter changes. The maximum amplitude frequency values at PSD signs for LFP activities of cell groups have been evaluated for maximum frequency. See section 2 for information on PSD and SR calculations.

Figure 8:

Maximum frequency, PSD, and SR changes in the cortical network. Maximum frequency (Hz), PSD (dB/Hz), and SR change graphs calculated based on and parameter changes. The maximum amplitude frequency values at PSD signs for LFP activities of cell groups have been evaluated for maximum frequency. See section 2 for information on PSD and SR calculations.

Figure 9:

Average maximum SR and PSD changes in the cortical network. The average curves are obtained, respectively, by the average horizontal (based on and vertical (based on axis of the graphs in Figure 8. The average maximum SR and maximum PSD (inset) curves (a) for 11 different values depend on parameter and (b) for 11 different parameters depending on the parameter of cortical cell groups.

Figure 9:

Average maximum SR and PSD changes in the cortical network. The average curves are obtained, respectively, by the average horizontal (based on and vertical (based on axis of the graphs in Figure 8. The average maximum SR and maximum PSD (inset) curves (a) for 11 different values depend on parameter and (b) for 11 different parameters depending on the parameter of cortical cell groups.

Figure 10a shows the spike activities of the PY and IN cell groups in the cortical network in the format of raster graphics for the and value pairs. Although all the initial conditions and the external inputs are the same, the spike activity pattern of the network is significantly different depending on the and values. Moreover, although the amplitudes of the cortical cell groups are different for all and values, they exhibit oscillations at frequencies near the input frequency (marked with an arrow in Figure 10b). Although the external input activity is the same, for the value pair , where is below the threshold, the cortical PY and IN cells exhibit relatively low SR values and weak spike activities with low input sensitivity, as shown in Table 7. However, for and value pairs, the SR value is relatively higher, so the input sensitivities of the cortical PY and IN groups are higher, as seen in the raster and LFP graphs and they exhibit a stronger oscillating activity consistent with the periodic input activity (see Table 7).

Figure 10:

Raster, LFP, and PSD graphs for various and value pairs in the cortical network. (a) Raster graphs and (b) LFP and spectral analyses of cortical cell groups are presented for and value pairs used in Figure 4. Under the same external input (red), the spike activities for cortical PY (black) and IN (blue) cell groups are shown in the raster graphs. The dominant maximum amplitude frequency component in the PSD signs of total LFPs of cell groups is indicated with a black arrow.

Figure 10:

Raster, LFP, and PSD graphs for various and value pairs in the cortical network. (a) Raster graphs and (b) LFP and spectral analyses of cortical cell groups are presented for and value pairs used in Figure 4. Under the same external input (red), the spike activities for cortical PY (black) and IN (blue) cell groups are shown in the raster graphs. The dominant maximum amplitude frequency component in the PSD signs of total LFPs of cell groups is indicated with a black arrow.

According to the analyses based on spike times and the results of spectral analyses of LFP signs for the cortical PY and IN cell groups, although the input activity and initial conditions are the same, it is observed that at high and values, both input sensitivity and spike coherence are increased. This shows that the subthreshold rhythmic activities present in the cortical cell characteristics resonate at suitable and values and regulate the cellular activity and the spike activity in the overall network.

The generalization rate of the method is figured in terms of multiple simulations. The mean square errors (MSE) obtained for the studied parameters of the network are given in Table 6. The MSE values were calculated and averaged over ) trials, where each simulation was taking almost 3.5 days.

Table 6:
MSE Errors for Analysis.
AnalysisPYIN
Number of spike 0.06478 0.02116 
Frequency 0.05597 0.02062 
Spike Coherence 0.08888 0.01705 
StochasticResonance 0.06678 0.05857 
MaxFrequencies 0.00395 0.00404 
MaxPSDs 0.01041 0.00963 
AnalysisPYIN
Number of spike 0.06478 0.02116 
Frequency 0.05597 0.02062 
Spike Coherence 0.08888 0.01705 
StochasticResonance 0.06678 0.05857 
MaxFrequencies 0.00395 0.00404 
MaxPSDs 0.01041 0.00963 

4  Discussion

In many nonlinear systems, the synergy roaming in the neuronal system somehow bring about physiological and bioelectrical rhythmic activities with essential signals that play important roles in living functional biology. The sources of these rhythmic activities mutually interact each other and the environment. Various factors such as the topology of the system and synaptic background noise (and many other factors not considered here) are very effective on these rhythmic activities. In the neuronal system/brain, the spatiotemporal format of spiky discharges of neurons is thought to reveal important information regarding brain function. Slow waves like electrical oscillations resulting from synchronized spiking activity of large-scale neuronal networks have been analyzed in a variety of perspectives to understand the functional and physiological structure of the brain.

It is thought that for each cognitive process in the brain, the neuronal network swaps into clusters in relation to information being processed (Varela et al., 2001). For example, neural groups that exhibit specialized and synchronized activity for each recognized object make it possible to see different visual properties such as edges, color, texture, and depth in different visual regions (Roskies, 1999). This parallel phase synchronization between different spectral bands is said to differentiate structures (Bressler, Coppola, & Nakamura, 1993; Fries, Reynolds, Rorie, & Desimone, 2001; Von Stein, Chiang, & König, 2000; Von Stein, Rappelsberger, Sarnthein, & Petsche, 1999). This parallel phase synchronization between different bands allows integrated processes between different brain regions (far and/or near regions). Moreover, that characteristic frequency bands emerge in different individuals during the same task or behavior proves their significance in cognition mechanism (Varela et al., 2001). It seems possible to say that cognitive experiences acquired during wakefulness are consolidated during sleep, and the neuronal network engenders some slow oscillations that could carry meaningful information related to various high-frequency activities processed in both far and near brain regions. The studies conducted on the functional structure of brain networks show that a cortical network resembles synthetically structured small world networks in being able to switch between randomness and regular structures with respect to the function taken on (Hadley et al., 2016). With variation of some parameters, the neuronal network can move from a regular/grouping state (wakefulness) to a much randomized state (sleep or anesthesia). Similarly the SW network can move between a clustering state () and a randomized state (). For the network exhibits localized integrative short paths between neighboring cells, while for , the network exhibits short-cuts between far-range cell groups, similar to the brain’s characteristic behavior.

This study was conducted to investigate the relationship between slow rhythmic electrical oscillations, which mainly emerge in the thalamus and distribute over a wide range of the thalamocortical region, particularly during sleep and anesthesia, using the designed SW topological cortical network. The SW was designed for modeling an isolated cortical network to understand the role of and in the cortex exclusively on these slow rhythmic activities. For all and values, the SW outputs were calculated with the same initial conditions and external inputs considered as coming from the thalamus. This work did not in fact involve the thalamus, but it can be noted that the thalamus is the origin of such activities. The information that we have come across through invivo (Deschênes, Paradis, Roy, & Steriade, 1984; Steriade & Deschenes, 1984; Timofeev & Steriade, 1996) and in vitro (Bal, Von Krosigk, & McCormick, 1995a, 1995b, 1995c; Von Krosigk, Bal, & McCormick, 1993; Von Krosigk, Monckton, Reiner, & McCormick, 1999) studies reveals that slow waves commonly originate in the thalamus and later spread over much a larger scale of thalamic and cortical regions and that the cortex holds a critical role in synchronizing these slow waves (Contreras et al., 1996; Contreras & Steriade, 1996a, 1996b; Contreras, Timofeev, & Steriade, 1996; Steriade, 2001). Bishop (1936) showed that slow waves are generated within the thalamus, and they are terminated as the cortex is isolated from the thalamus. Therefore, the cortico-thalamic feedbacks are thought to be the basic mechanism underlying the synchronization of RE neuronal cells and, consequently causing the formation and amplification of thalamic spindles. These spindles may sometimes enter into a spatiotemporally coherent phase (Contreras et al., 1997; Contreras, Destexhe et al., 1996; Contreras, Timofeev et al., 1996; Destexhe, Contreras et al., 1998). On the contrary, some other studies revealed that the feedbacks between thalamic and cortical mechanisms also stop the spindles (Bonjean et al., 2011). Also it has been stated in detail that slow waves start in local fields as a consequence of the synchronized discharges arising with respect to the neuronal excitatory and inhibitory correlation between TC and RE cells in that region (Beenhakker & Huguenard, 2009).  Later, with the effect of cortical projections of this small group of cells, cortical TC and RE cells discharge and burst, cropping up and spreading over a wide range of the cortical field in the form of spindles. As the cells that cause the spindle waves are synchronously excited, they influence neocortical cells and cause these waves to synchronize on a large scale in the cortex (Amzica & Steriade, 1997). The sleep spindles are confirmed to start in the deep brain in thalamic reticular nucleuses and synchronize through cortico-thalamic and thalamocortical closed loops (Steriade, 2006), which also play or amplification role.

It has also been acknowledged that the thalamus is responsible for organizing the frequency content of cortical EEG and, like a major gate, plays an important role in conducting sensorial signals to the cortex (Evans, 2007). Rodolfo Llinas declared that rather than being a gate, the thalamus behaves similar to a central hub in establishing communication links with other brain regions (Evans, 2007). Unfortunately we did not come across a synthetically designed network such as SW in investigating the parametric effects of the thalamic or thalamo-cortical network on these slow waves. But it is clear that cortical inputs besides synchronization amplify these waves all the way through TC neurons and that cortico-thalamic stimuli are much more effective than prethalamic stimuli (Contreras & Steriade, 1996a).

Through the results obtained in this study, it was observed that the and parameters affect spatial synchronization and temporal periodicity. At some and values, all cell groups in the SW cortical network produced coherent spike discharges among themselves and with other cell groups and, accordingly, exhibited strong synchronized oscillatory waves. To analyze and quantify the characteristics of these waves, the ranges of synchronization, coherency, and statistical means such as SR and PSD were calculated.

It was observed that almost all of the cortical PY cells were insensitive to input and did not create a valuable range of spike activity for . As gradually increased (, many spatiotemporally coherent spiky discharges with a reasonably high PSD (around 28 in average; see Figure 9) took place with respect to the input in PY neurons in the SW cortical network. In cortical IN cells, the spike discharge rate was somewhat less, and the as value increased, the input sensitivity of cortical cell groups increased along with their activities and reached maximum values around . At values higher than 0.04. As the