Generation of theta oscillations by weakly coupled neural oscillators in the presence of noise

Abstract

Neuronal oscillations are a robust phenomenon occurring in a variety of brain regions despite considerable amounts of noise. In this article classical phase-response theory is generalized to the case of noisy weak-coupling regimes by deriving an iterated map for the asynchrony of spikes in an oscillation cycle. Two criteria are introduced to check the validity of our approximations: One criterion tests the assumption that all neurons fire exactly once per cycle, the other criterion tests for linearity. The framework is applied to stellate cells of the medial entorhinal cortex layer II. We find that rhythmogenesis is more robust in the case of excitatory noise as compared to inhibitory noise. It is shown that a network of stellate cells can also act as a generator of theta if the neurons are connected via a fast-oscillating network of inhibitory interneurons.

A Appendix

1.1 A.1 Parameters of numerical simulations

We simulate a network of 500 all-to-all synaptically-coupled neurons. All simulations were done on a Linux-cluster using C++ and LAM-MPI. The integration method was an adaptive fourth-order Runge-Kutta method (Press et al., 1992).

1.1.1 A.1.1 Stellate cell model

The neuron model used in this paper has been introduced by Acker et al. (2003) for stellate cells of layer II of the medial entorhinal cortex. With the membrane capacitance C and the membrane potential $V_m$ (in mV), the current-balance equation

$$ C\frac{{\rm d} V_m}{{\rm d}t}= I_{\rm DC} - I_\eta - I_{\rm Na}-I_{\rm Nap} - I_{\rm K} - I_{\rm H}- I_{\rm L}-I_{\rm syn} $$

(26)

contains active conductances mediating a rectifying potassium current $I_{\rm K}$, an unspecific cation current $I_{\rm H}$ as well as fast and persistent sodium currents $I_{\rm Na}$ and $I_{\rm Nap}$, respectively. In addition, the model includes a passive leak current $I_{\rm L}$, a constant current $I_{\rm DC}$, a synaptic current $I_{\rm syn}$, and a noise current $I_\eta$. With $g_x$ denoting the voltage-dependent conductance of the x-th ionic current and $V_x$ its reversal potential, the above currents are modeled as

$$ I_{\rm Na} = g_{\rm Na}\;m_{\rm Na}^3\;h_{\rm Na}\;(V_m-V_{\rm Na})$$

(27)

$$ I_{\rm Nap} = g_{\rm Nap}\;m_{\rm Nap}(V_m-V_{\rm Na}) $$

(28)

$$ I_K = g_K\; n_K^4(V_m-V_K) $$

(29)

$$ I_H = g_H\;(0.65\;m_{\it f}+0.35\;m_{\it s})\; (V_m-V_H)$$

(30)

$$ I_L = g_L\;(V_m-V_L) $$

(31)

The noise current $I_\eta = g_\eta(V_m-V_\eta)$ is generated by a stochastic conductance $g_\eta$ that is modeled by a Gaussian process η with zero mean and standard deviation $\sigma_{\rm \eta}$ that is independent and identically distributed across the population of neurons. The synaptic current is specified below (A.1.2).

The dynamics of the activation and inactivation variables, $m_x,\,h_x\, {\rm and}\,n_x$, obey first-order kinetics and is usually expressed as

$$ \frac{{\rm d}x}{{\rm d}t} =\alpha_x(V_m) (1-x)-\beta_x(V_m) x $$

(32)

where $\alpha_x$ and $\beta_x$ are forward and backward rates, respectively, governing the transitions of the channels between open and closed states. In the case of $I_h$ we use the description

$$ \frac{{\rm d}x}{{\rm d}t}=\frac{x_\infty(V_m)-x}{\tau_x(V_m)} $$

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where $x_\infty$ denotes the steady-state value of x and $\tau_x$ its time constant, both as functions of the membrane voltage.

In what follows all currents are measured in units of $\rm \mu A/cm^2$, all voltages are measured in mV. The unit of time is ms. Then, the forward and backward rate constants are given by

$$ \alpha_{\rm m,Na} = \left\{\begin{array}{ll} \displaystyle\frac{-0.1\;(V_m+23)}{\exp(-0.1\;(V_m+23))-1} & \quad{\rm for}\; V_m\neq -23 \nonumber\\ 1 & \quad{\rm for}\; V_m=-23 \end{array}\right.\\ $$

(34)

$$ \beta_{\rm m,Na} = 4 \exp\left(\frac{-(V_m+48)}{18}\right) $$

(35)

$$ \alpha_{\rm h,Na} = 0.07 \exp\left(\frac{-(V_m+37)}{20}\right) $$

(36)

$$ \beta_{\rm h,Na} = \frac{1}{\exp(-0.1\;(V_m+7))+1} $$

(37)

$$\alpha_{\rm m,Nap} = \frac{1}{0.15\,(1+\exp(-(V_m+38)/6.5))} $$

(38)

$$\beta_{\rm m,Nap} = \frac{\exp(-(V_m+38)/6.5)}{0.15\,(1+\exp(-(V_m+38)/6.5))} $$

(39)

$$\alpha_{\rm n,K} = \left\{\begin{array}{ll} \displaystyle\frac{-0.01(V_m+27)}{\exp(-0.1(V_m+27))-1} &\quad {\rm for}\; V_m\neq -27 \nonumber\\ 1 &\quad {\rm for}\; V_m=-27 \end{array}\right.\\ $$

(40)

$$\beta_{\rm n,K} = 0.125\,\exp\left(\frac{-(V_m+37)}{80}\right). $$

(41)

The steady state values ($x_\infty$) and time constants $(\tau_x)$ are defined as

$$ m_{\rm f\infty, H} = \frac{1}{(1+\exp((V_m+79.2)/9.78))} $$

(42)

$$ \tau_{\rm m_f, H} = \frac{0.51}{\exp((V_m-1.7)/10)+\exp(-(V_m+340)/52)}+1 $$

(43)

$$m_{\rm s\infty, H} = \frac{1}{(1+\exp((V_m+71.3)/7.9))} $$

(44)

$$\tau_{\rm m_s, H} = \frac{5.6}{\exp((V_m-1.7)/14)+\exp(-(V_m+260)/43)}+1 $$

(45)

The model can exhibit both type I and type II oscillatory behavior, depending on whether the conductance of the h-current is switched on or off. The constant current $I_{\rm DC}$ was adapted such that the neuron oscillates with a frequency of 14 Hz. For type I, we use $g_h=0\;{\rm mS/cm^2}$, $I_{\rm DC}=1.4\;{\rm \mu A/cm^2}$. Type II is obtained with $g_h=1.5\;{\rm mS/cm^2}$, $I_{\rm DC}=-1.467\;{\rm \mu A/cm^2}$.

Other parameter values are as follows: $V_{\rm Na}$=55, $V_K$=–90, $V_h$=–20, $V_L$=−65; $g_{\rm Na}$=52 $\;{\rm mS/cm^2}$, $g_{\rm Nap}$=0.5 $\;{\rm mS/cm^2}, g_K$=11$\;{\rm mS/cm^2}$, $g_L$=0.5$\;{\rm mS/cm^2}$; C=1.5 $\;{\rm \mu F/cm^2}$.

1.1.2 A.1.2 Synapse model

The synaptic current

$$ I_{\rm syn} = g_{\rm syn}(t)\;(V_m-V_{\rm syn}) $$

(46)

is determined by the synaptic reversal potential $V_{\rm syn}$, and the conductance

$$ g_{\rm syn}(t) = \sum_{\rm spike} \kappa(t-t_{\rm spike}) $$

(47)

in which the sum runs over all the spikes emitted by the presynaptic neurons at times $t_{\rm spike}$. The time course of the synaptic conductance is modeled by an exponential decay

$$ \kappa(t)= {\cal H}(t-\Delta) g_{\rm syn}\,\exp[-(t-\Delta)/\tau_{\rm syn}] $$

(48)

with time constant,$\tau_{\rm syn} = 5$ ms. Here,${\cal H}(x)$ denotes the Heaviside step function that equals 1 for $x\ge0$ and is zero otherwise. The maximal synaptic conductance $g_{\rm syn}$ takes values from 5$\cdot 10^{-6}$ to 15$\cdot 10^{-6}\;{\rm mS/cm^2}$. The reversal potentials are taken to be $V_{\rm syn}^{\rm exc}$=0 mV, $V_{\rm syn}^{\rm inh}$=–80 mV.

1.2 A.2 Phase-response curves of the stellate cell

The excitatory and inhibitory experimental PRCs of the stellate cell were taken from the first and second panels of Fig. 4(A) in Netoff et al. (2005). There, the authors measured the synaptic PRCs from a entorhinal stellate cell by dynamic clamp recordings. PRCs were generated by delivering artificial excitatory or inhibitory conductance inputs to a stellate cell that has been firing periodically with a frequency of 10 Hz. We use PRCs induced by perturbation conductance amplitudes of 1 nS and 562 pS (solid lines in Fig. 4(A) of Netoff et al. (2005)) which is reported to correspond approximately to two to five times the size of spontaneous excitatory synaptic inputs to stellate cells (Berretta and Jones, 1996).