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Methods

The network model we considered is a model of synaptically connected neurons generating action potentials. Neurons are modeled as single compartment units using a conductance based model. The 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 overall strength of a connection is represented by a single synaptic weight parameter and a delay parameter represents all delays between cells. We use a two dimensional array of up to 250x250 neurons to simulate a two dimensional neural network (e.g. a thin slice or layer of neocortical tissue). Two types of networks were simulated. In the first type there is no significant inhibition. This can be interpreted either as a local network after applying a blocker of inhibition or cortical tissue where local inhibition is dominated by excitation. The second type of network has inhibitory neurons that are uniformly distributed with one inhibitory neuron occurring per nine excitatory neurons. Each neuron receives excitatory input from two of the nearest eight neighboring cells (Fig. 1). All connections have equal strength and delay. A pseudo-random generator was used to choose connections for each cell. The selected neurons (1-3) in the center of array are stimulated by a rectangular pulse (30 ms) train of depolarizing current with the frequency in the range 2 -10 Hz at the beginning of simulation. The membrane potentials for selected neurons and counts of action potentials for all neurons were determined. The data were later used to generate animations of the activity in neuronal arrays. The membrane potentials for selected border neurons were used to compute the frequency of bursts.

Neuron model equations

\begin{eqnarray*} C_m \frac{dV}{dt} & =& I_{syn}-I_{Na}-I_{Ca}-I_K-I_{K(Ca)}-I_... ...\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/... ... -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})\... ..._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 function constants.

Synaptic model equations

\begin{eqnarray*} I_{syn} &=& \sum_{j=1}^{N_{syn}}w_j g_j(t) (V-E_{syn})\\ g(... ...c{-\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 $E_{syn}$ is a synaptic reversal potential equal $-10~mV$ for excitatory and $-120~mV$ for inhibitory synapse. $\tau_d=3~ms$ and $\tau_o=0.5~ms$ represent respectively decay time and onset time constants of a PSP. Synaptic weight $w_j$ was modeled as an integer in the range $0~-~120$. $\Delta t_i$ 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, in these simulations $N_{syn}=4$ (2 excitatory and 2 inhibitory inputs). The ordinary differential equations were solved numerically using forward Euler method with a time step of 0.01 $msec$.
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Next: Figure 1 Up: index Previous: Introduction