Methods

The network model we considered is a model of synaptically connected reduced neurons generating action potentials. Cells are modeled as single compartment units using modified Av-Ron-Rinzel's reduced model equations. The neuron model incorporates two inward currents - $I_{Na}$ and $I_{Ca}$, three outward potassium currents - the delayed rectifier $I_{K}$, the Ca-dependent $I_{K(Ca)}$ and the transient $I_{A}$ current, and a leak current $I_{L}$. The synaptic connection between cells is modeled by a synaptic current $I_{syn}$. The synaptic conductance is represented by a sum of two exponential functions. The overall strength of a connection is represented by a single synaptic weight parameter and a delay parameter represents all delays between cells. In these simulations we use a network of 81 excitatory cells and 9 inhibitory cells to simulate a small locally connected region of the brain tissue (Fig. 1). Presynaptic neurons are chosen randomly from all 81 excitatory and 9 inhibitory neurons respectively. A pseudo-random generator was used to choose connections for each cell. This produced a network with no predefined structure of circuits. Each neuron has preset number of excitatory inhibitory connections same for all neurons in a given simulation. Number of excitatory connections on input was 2,3 or 4 and number of inhibitory connections was 3,5 or 6 respectively. The weight of synaptic connections is equal to 60 for excitatory connections and 120 for inhibitory connections. The delay is $3.6 \pm 0.5 ms$. Individual cells and synapses have properties based on physiologic data. The network was activated by applying random (Poisson) excitatory input to 4 or 9 selected cells.

To compare results of simulation for different networks we use a measure of the degree of synchrony in a network $\Delta$. To calculate this measure of synchrony we follow the methods proposed by (Hansel et al., 92, Hansel et al., 98). This method is based on analysis of temporal fluctuations of the voltage activity of all neurons in the network. First at a given time $t$ the average membrane potential of all neurons in the network is calculated and next the variance of the average membrane potential over a time period $<t>$ is computed.

$$\begin{eqnarray*} {\Delta}_N &=& \langle { \left ( \frac{1}{N} \sum_{i=1}^{N} V_... ...{1}{N} \sum_{i=1}^{N} V_i(t)} {\rangle}_t \right)^{\frac{1}{2}} \end{eqnarray*}$$

In the same way the average variance of the voltage activity of a single neuro is calculated over all neurons in the network.

$$\begin{eqnarray*} {\Delta} &=& \frac{1}{N} \sum_{i=1}^N \left( \langle {V_i(t)}^... ...rangle}_t - {\langle V_i(t) {\rangle}_t}^{\frac{1}{2}} \right) \end{eqnarray*}$$

The variance of the average membrane potential in the network over time period $<t>$ is normalized to the average (in neuron population) variance of the activity of a single cell over the same time period. For a large N (number of neurons), this gives a value of $\Delta$ between 0 and 1. The resulting parameter describes a degree of synchrony (coherence) in the network in the finite time period (0 means the network is in an asynchronous state, 1 means neuronal activity in the network is total synchronized). In this work the computation of the measure of degree of synchrony is performed on neuron voltage outputs with sampling frequency 10 kHz over $t = 5 sec$ of simulation with $N = 90$

Neuron model equations

$$\begin{eqnarray*} C_m \frac{dV}{dt} & =& I_{syn}-I_{Na}-I_{Ca}-I_K-I_{K(Ca)}-I_A... ..._{\infty}(V)-B}{\tau_{_B}},\\ \frac{dC}{dt} &=& K_p(-I_{Ca})-RC \end{eqnarray*}$$

where:

$$\begin{eqnarray*} \tau_{_W}(V) &=& \frac {1}{\lambda}\left(e^{a^{(W)} (V-V_{1/2}... ...^{ -2a^{(P)}(V-{V_{1/2}^{(P)}})}\right)^{-1},  for P=W,m,X,A,B, \end{eqnarray*}$$

$V$ is the membrane potential, $W$ is the recovery variable, $C$ is the intracellular calcium concentration $X$ and $B$ are respectively the calcium channel activation variable and transient potassium channel inactivation variable. The steady-state functions $m_{\infty}$, $A_{\infty}$, $W_{\infty}$, $X_{\infty}$, and $B_{\infty}$ are modeled as sigmoidal curves, determined by two parameters: the half maximum voltage $V_{1/2}$(values are -31, -20, -35, -45 and -70 $mV$ respectively) and a slope $a$ of the curve at this point (values are 0.065, 0.02, 0.055, 2.0, and -0.095 respectively). $K_p=0.0002$ is the conversion factor from calcium current to concentration and $R=0.006$ is the removal rate constant of the intracellular calcium concentration. $C_m=1 \mu F/cm^2$ is the membrane capacitance. $\tau_{_W}$ is the relaxation time function, and $\tau_{_X}=25 msec$ and $\tau_{_B}=10 msec$ are relaxation time constants for recovery $W$, calcium activation $X$, and potassium transients inactivation $B$ variables. Ion currents $I_i$ are described by the product of three terms: the maximal conductance $\overline{g}_i$, the activation and inactivation variable or function, and the driving force $(V-V_i)$.

$$\begin{eqnarray*} I_{Na} & = & \overline{g}_{Na}m_{\infty}^3(V)(1-W)(V-V_{Na})\\... ...g}_A A_{\infty}(V)B(V-V_K)\\ I_L & = & \overline{g}_L(V-V_L)\\ \end{eqnarray*}$$

where: $\bar{g}_{Na}=120  mS/cm^2$, $\bar{g}_{Ca}=1.0  mS/cm^2$, $\bar{g}_K=15  mS/cm^2$, $\bar{g}_A=12.5  mS/cm^2$, $\bar{g}_L=0.3  mS/cm^2$, $\bar{g}_{K(Ca)}$ in the range 0.5-3.5 $mS/cm^2$ are maximum conductances for the respective channels and $V_{Na}=-50 mV$, $V_{Ca}=124 mV$, $V_{K}=-72 mV$, and $V_L=-50 mV$ are values of the reversal potentials for the respective ions and leak current. $K_d=0.5$ and $K_C=2$ are the calcium concentration functions constants.

Synaptic model equations

$$\begin{eqnarray*} I_{syn} &=& \sum_{j=1}^{N_{syn}}w_j g_j(t) (V-E_{syn})\\ g(t)... ...rac{-\Delta t_i}{\tau_d}}-e^{\frac{-\Delta t_i}{\tau_o}}\right) \end{eqnarray*}$$

where $i$ denotes summation over past action potentials and $j$ over the number of input synapses. $\overline{g}_{syn}=0.0112 mS/cm^2$, is the synaptic conductance constant, $g_j(t)$ is the conductance function and $E_{syn}=-10 mV$ is a synaptic reversal potential. $\tau_d=3 ms$ and $\tau_o=0.5 ms$ represent respectively decay time and onset time constants of a PSP. Synaptic weight $w_j$ was modeled as integer 60 for excitatory connections and -120 for inhibitory connections. $\Delta t_i$ denotes time elapsed since $i$-th action potential arrival on synapse, $N$ is the number of past action potentials with significant contribution to the sum and $N_{syn}$ is the number of synaptic inputs.

The ordinary differential equations were solved numerically using the forward Euler method with a time step of 0.01 $msec$.

The network is constantly activated by random (Poisson) excitatory input applied to selected neurons.