GABA diffusion across neuronal columns for efficient sensory tuning

Abstract

Synaptic (phasic) lateral inhibition between neuronal columns mediated by GABAergic interneurons is, in general, essential for primary sensory cortices to respond selectively to elemental features. We propose here a neural network model with a nonsynaptic (tonic) lateral inhibitory mechanism. While firing, intrasynaptic GABA molecules spill over into extracellular space and accumulate in neuronal columns. Through accumulation in and diffusion across these columns, a level of ambient (extracellular) GABA changes in a neuronal activity-dependent manner. Ambient GABA molecules act on extrasynaptic receptors and provide neurons with tonic inhibitory currents. We examined whether and how the diffusion of GABA molecules across neuronal columns affects tuning performance of the network to a feature stimulus: selective responsiveness. The GABA diffusion led to reducing ambient GABA in the stimulus-relevant column while augmenting ambient GABA in stimulus-irrelevant columns, thereby improving the tuning performance. The GABA diffusion was effective especially when provided with a broader sensory input. Interestingly, this diffusion-based, nonsynaptic (tonic) lateral inhibitory scheme worked well together with the conventional, synaptic (phasic) lateral inhibitory scheme, enhancing the sensory tuning. We suggest that the nonsynaptic lateral inhibition, mediated through GABA diffusion across neuronal columns, may be beneficial for the cortex to tune to sensory features.

Appendices

Appendix 1: Description of network model

Dynamic evolution of membrane potential of the ith P cell belonging to column n is defined by

$$\begin{aligned} c_m^{P} \frac{\hbox {d}u_{i}^{P}(n;t)}{dt}= & {} -g_m^{P}(u_{i}^{P}(n;t)-u_\mathrm{rest}^{P}) + I_{i,\mathrm{rec}}^{P}(n;t) \nonumber \\&+\, I_{i,\mathrm{fed}}^{P}(n;t) + I_{i,\mathrm{lat}}^{P}(n;t) + I_{i,\mathrm{ext}}^{P}(n;t) \nonumber \\&+\, I_\mathrm{inp}^P(n), \end{aligned}$$

(6)

where $I_{i,\mathrm{rec}}^{P}(n;t)$ is a recurrent excitatory synaptic current from other P cells within the same column. $I_{i,\mathrm{fed}}^{P}(n;t)$ is a feedback inhibitory synaptic current from an F cell, $I_{i,\mathrm{lat}}^{P}(n;t)$ a lateral inhibitory synaptic current from L cells, $I_{i,\mathrm{ext}}^{P}(n;t)$ an inhibitory nonsynaptic current mediated by ambient GABA via extrasynaptic receptors and $I_\mathrm{inp}^P(n)$ an excitatory current triggered by an input stimulus: $f_\mathrm{inp}$ where $inp \in \{0, 1, 2,\ldots , n,\ldots ,M\}$. These currents are defined by

$$\begin{aligned}&I_{i,\mathrm{rec}}^{P}(n;t) = -\hat{g}_\mathrm{AMPA}\left( u_i^{P}(n;t) - u_\mathrm{rev}^\mathrm{AMPA}\right) \nonumber \\&\qquad \times \sum _{j=1}^Nw_{ij,\mathrm{rec}}^{P}(n)r_{j}^{P}(n;t),\end{aligned}$$

(7)

$$\begin{aligned}&I_{i,\mathrm{fed}}^{P}(n;t)= -\hat{g}_\mathrm{GABA}\left( u_i^{P}(n;t) - u_\mathrm{rev}^{GABA}\right) w_{i,\mathrm{fed}}^{P}(n)r_{i}^F(n{;}t),\end{aligned}$$

(8)

$$\begin{aligned}&I_{i,\mathrm{lat}}^{P}(n;t) = -\hat{g}_\mathrm{GABA}\left( u_i^{P}(n;t) - u_\mathrm{rev}^{GABA}\right) \nonumber \\&\qquad \times \sum _{j=1}^Nw_{ij,\mathrm{lat}}^{P}(n)r_{j}^{L}(n;t),\end{aligned}$$

(9)

$$\begin{aligned}&I_{i,\mathrm{ext}}^P(n;t) = -\hat{g}_\mathrm{GABA}\left( u_i^{P}(n;t) - u_\mathrm{rev}^{GABA}\right) \delta _P r_\mathrm{ext}^{GABA}(n;t),\end{aligned}$$

(10)

$$\begin{aligned}&I_\mathrm{inp}^P(n) = \alpha _P \hbox {e}^{-\left( \frac{n-inp}{\tau _\mathrm{inp}}\right) ^2}. \end{aligned}$$

(11)

Dynamic evolution of membrane potentials of the ith F and L cells is defined by

$$\begin{aligned} c_m^{F} \frac{\hbox {d}u_{i}^F(n;t)}{dt}= & {} -g_m^F(u_{i}^F(n;t)-u_\mathrm{rest}^F) \nonumber \\&+\, I_i^F(n;t) + I_{i,\mathrm{ext}}^F(n;t),\end{aligned}$$

(12)

$$\begin{aligned} c_m^{L} \frac{\hbox {d}u_{i}^{L}(n;t)}{dt}= & {} -g_m^{L}(u_{i}^{L}(n;t)-u_\mathrm{rest}^{L}) \nonumber \\&+\, I_i^{L}(n;t) + I_{i,\mathrm{ext}}^L(n;t), \end{aligned}$$

(13)

where $I_i^F(n;t)$ and $I_i^{L}(n;t)$ are excitatory synaptic currents from P cells. $I_{i,\mathrm{ext}}^F(n;t)$ and $I_{i,\mathrm{ext}}^L(n;t)$ are inhibitory nonsynaptic currents. These currents are defined by

$$\begin{aligned}&I_i^F(n;t) = -\hat{g}_\mathrm{AMPA}(u_i^F(n;t) - u_\mathrm{rev}^\mathrm{AMPA})w_i^F(n)r_{i}^P(n;t),\end{aligned}$$

(14)

$$\begin{aligned}&I_{i,\mathrm{ext}}^F(n;t) = -\hat{g}_\mathrm{GABA}(u_i^F(n;t) - u_\mathrm{rev}^{GABA}) \delta _F r_\mathrm{ext}^{GABA}(n;t),\end{aligned}$$

(15)

$$\begin{aligned}&I_i^{L}(n;t)= -\hat{g}_\mathrm{AMPA}(u_i^{L}(n;t) - u_\mathrm{rev}^\mathrm{AMPA}) \nonumber \\&\quad \quad \times \sum _{n'=0(n'\ne n)}^M w_{i}^L(n,n')r_{i}^{P}(n';t),\end{aligned}$$

(16)

$$\begin{aligned}&w_i^L(n,n') = W_L \hbox {e}^{-\left( \frac{n-n'}{\tau _\mathrm{lat}}\right) ^2},\end{aligned}$$

(17)

$$\begin{aligned}&I_{i,\mathrm{ext}}^L(n;t)= -\hat{g}_\mathrm{GABA}(u_i^L(n;t) - u_\mathrm{rev}^{GABA}) \delta _L r_\mathrm{ext}^{GABA}(n;t).\nonumber \\ \end{aligned}$$

(18)

In these equations, $r_j^P(n;t)$ is the fraction of AMPA receptors in the open state triggered by presynaptic action potentials of the jth P cell. $r_j^F(n;t)$ and $r_j^L(n;t)$ are the fractions of intrasynaptic GABAa receptors in the open state triggered, respectively, by presynaptic action potentials of the jth F and L cells. $r_\mathrm{ext}^{GABA}(n;t)$ is the fraction of extrasynaptic GABAa receptors in the open state provoked by ambient GABA in column n. $\delta _P$, $\delta _F$ and $\delta _L$ denote the amounts of extrasynaptic GABAa receptors on P, F and L cells, respectively. For model parameters and their values, see Table 1 and our previous studies (Hoshino 2007a, b, 2008b, 2011a; Totoki et al. 2010; Miyamoto et al. 2012). The receptor dynamics is defined in Appendix 2.

Appendix 2: Receptor dynamics

Receptor dynamics is based on a study (Destexhe et al. 1998) and described as

$$\begin{aligned}&\frac{\hbox {d}r_j^P(n;t)}{dt}= \alpha _\mathrm{AMPA}[\hbox {Glut}]_j(n;t)(1-r_j^P(n;t)) \nonumber \\&\qquad -\, \beta _\mathrm{AMPA} r_j^P(n;t),\end{aligned}$$

(19)

$$\begin{aligned}&\frac{\hbox {d}r_j^Y(n;t)}{dt}= \alpha _\mathrm{GABA}[\hbox {GABA}]_j^Y(n;t)(1-r_j^Y(n;t)) \nonumber \\&\qquad -\, \beta _\mathrm{GABA} r_j^Y(n;t), \quad (Y = F, L)\end{aligned}$$

(20)

$$\begin{aligned}&\frac{\hbox {d}r_\mathrm{ext}^{GABA}(n;t)}{dt}= \alpha _\mathrm{GABA}[\hbox {GABA}]_\mathrm{ext}(n;t)(1-r_\mathrm{ext}^{GABA}(n;t)) \nonumber \\&\qquad \qquad -\, \beta _\mathrm{GABA} r_\mathrm{ext}^{GABA}(n;t), \end{aligned}$$

(21)

where $[\hbox {Glut}]_j(n;t)$ and $[\hbox {GABA}]_j^Y(n;t)$ are concentrations of glutamate and GABA in the synaptic cleft, respectively. $[\hbox {Glut}]_j(n;t) = \hbox {Glut}_\mathrm{syn}$ and $[\hbox {GABA}]_j^Y(n;t) = \hbox {GABA}_\mathrm{syn}^Y$ for 1 ms when the presynaptic jth P cell and type Y cell fire, respectively. Otherwise, $[\hbox {Glut}]_j(n;t) = 0$ and $[\hbox {GABA}]_j^Y(n;t) = 0$. $[\hbox {GABA}]_\mathrm{ext}(n;t)$ is ambient (extracellular) GABA concentration in column n at $\hbox {time} = \hbox {t}$ [see Eq. (1)]. For model parameters and their values, see Table 1 and our previous studies (Hoshino 2008a, 2009, 2010, 2011b, 2012, 2013, 2015).

Appendix 3: Action potential generation and simulation method

Probability of firing of the jth Y cell belonging to column n is defined by

$$\begin{aligned} \hbox {Prob}[Y_j(n;t); firing]= & {} \frac{1}{1+\hbox {e}^{-\eta _Y(u_j^Y(n;t)-\zeta _Y)}},\nonumber \\&(Y = P, F, L) \end{aligned}$$

(22)

where $\eta _Y$ and $\zeta _Y$ are, respectively, the steepness and the threshold of the sigmoid function. When a cell fires, its membrane potential is depolarized to $-10\,\hbox {mV}$, which is kept for 1 ms and then reset to the resting potential. For model parameters and their values, see Table 1.

Time step for the calculation is 1 ms. C language is used for the numerical calculation and data analysis. It runs in a Windows PC with 10,000 iterations per 10 s simulation. We have confirmed that physiologically reliable membrane potentials, action potentials and ambient GABA concentrations can be obtained for time step less than 2 ms.