Spike timing precision of neuronal circuits

Abstract

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.

Appendix: : ion channel models

Neurons used in our numerical analyses include 18 K⁺ channels/μm² and 60 Na⁺ VG channels/μm² (Dayan and Abbott 2001). In analyzing various neuronal architectures, the number of excitatory and inhibitory synaptic receptor LG channels per synapse are varied, in accordance with the physiologically plausible values for the receptor channel densities, the number of synapses, and the membrane areas of the neurons (Masugi-Tokita et al. 2007; Chiu et al. 2002). We utilize well established kinetic schemes for K⁺ and Na⁺ channels, as shown in Fig. 14a and b (Dayan and Abbott 2001). Excitatory synapses include AMPA/kainate receptor channels with a kinetic scheme shown in Fig. 14c (Destexhe et al. 1998b). Inhibitory synapses include GABA _A receptor channels having a kinetic scheme illustrated in Fig. 14d (Destexhe et al. 1998b).

Fig. 14

MC models for a K⁺, b Na⁺, c excitatory AMPA/kainate receptor, and d inhibitory GABA _A receptor ion channels

The transition rates we use for the kinetic models of K⁺ and Na⁺ channels are given by

$$ \begin{array}{llllllllll} &\alpha_{n}(V(t))=\frac{0.01(V(t)+ 55)}{1-\exp[-(V(t)+ 55)/10]},\\ &\beta_{n}(V(t))= 0.125\exp[-(V(t)+ 65)/80],\\ &\alpha_{m}(V(t))=\frac{0.1(V(t)+ 40)}{1-\exp[-(V(t)+ 40)/10]},\\ &\beta_{m}(V(t))= 4\exp[-(V(t)+ 65)/18],\\ &\alpha_{h}(V(t))= 0.07\exp[-(V(t)+ 65)/20],\\ &\beta_{h}(V(t))=\frac{1}{1+\exp[-(V(t)+ 35)/10]}, \end{array} $$

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where V (t) is the membrane potential of the neuron as described by Eq. (1) and expressed in mV, α _i ’s and β _i ’s are expressed in msec^− 1 (Dayan and Abbott 2001). The ionic K⁺ and Na⁺ currents are given by

$$ \begin{array}{llllllllll} &I_{\mathrm{K}^{+}}=g_{\mathrm{K}^{+}}N_{n^{4}}[V(t)-E_{\mathrm{K}^{+}}],\\ &I_{\text{Na}^{+}}=g_{\text{Na}^{+}}N_{m^{3}h^{1}}[V(t)-E_{\text{Na}^{+}}], \end{array} $$

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where $g_{\mathrm {K}^{+}}=g_{\text {Na}^{+}}= 20$ pS, $E_{\mathrm {K}^{+}}=-77$ mV, $E_{\text {Na}^{+}}= 50$ mV, $N_{n^{4}}$ and $N_{m^{3}h^{1}}$ are the total number of open K⁺ and Na⁺ channels, respectively (Dayan and Abbott 2001).

Nominal values for the transition rates in the kinetic model of the AMPA/kainate receptor channel, i.e., $R_{b}([L(t)])= 1.3\times 10^{7}[L(t)], R_{u_{1}}= 5.9\times 10^{0}, R_{u_{2}}= 8.6\times 10^{4}, R_{r}= 6.4\times 10^{1}, R_{d}= 9.0\times 10^{2},R_{c}= 2.0\times 10^{2}$, and R _o = 2.7 × 10³ (all in units of s^− 1), are given in (Destexhe et al. 1998b). The results presented in Section 5 were obtained based on these transition rates for the AMPA/kainate receptor channel. With these values, the synaptic delay is approximately 1 msec. In order to increase the synaptic delay (between the spikes generated in the presynaptic and postsynaptic neuron connected via an excitatory synapse) by ∼1 msec to 2 msec, we also use updated transition rates for the AMPA/kainate receptor channels, given by $R_{b}([L(t)])= 1.0\times 10^{6}[L(t)], R_{u_{1}}= 3.0\times 10^{1}, R_{u_{2}}= 1.6\times 10^{5}, R_{r}= 3.4\times 10^{1}, R_{d}= 8.4\times 10^{2}, R_{c}= 3.7\times 10^{2}$, and R _o = 0.9 × 10³ (all in units of s^− 1). The results presented in Sections 4 and 6 were obtained based on these updated transition rates for the AMPA/kainate receptor channel. These updated values are justified based on the diversity of activation and desensitization properties of AMPA/kainate receptor channels that is described in Pinheiro and Mulle (2006) and Perrais et al. (2010). For neurons that are reciprocally connected by excitatory and inhibitory synapses, a synaptic delay that is shorter than the depolarization duration of an action potential would result in a premature inhibition of the action potential. Thus, a ∼1 msec increase in the synaptic delay makes the neuronal dynamics more physiologically meaningful. Prolonged synaptic delays can alternatively be modeled via distributed multi-compartment synapse or neuron models.

The transition rates in the kinetic model of the GABA _A receptor channel are set to $R_{b_{1}}([L(t)])= 2.0\times 10^{7}[L(t)], R_{b_{2}}([L(t)])= 1.0\times 10^{7}[L(t)], R_{u_{1}}= 4.6\times 10^{3}, R_{u_{2}}= 9.2\times 10^{3}, R_{c_{1}}= 9.8\times 10^{3}, R_{c_{2}}= 4.1\times 10^{2}, R_{o_{1}}= 3.3\times 10^{3}$, and $R_{o_{2}}= 1.1\times 10^{4}$ (all in units of s^− 1), as given in Destexhe et al. (1998b).

[L(t)] is the concentration of neurotransmitter molecules in the synaptic cleft, given by

$$ [L(t)]=\frac{L_{\max}}{1+\exp[-(V_{\text{pre}}(t)-V_{p})/K_{p}]} $$

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where L_max = 2.84 × 10^− 3 M is the maximal concentration, V_pre(t) is the membrane potential of the presynaptic neuron expressed in mV, K _p = 5 mV is the steepness parameter and V _p = 2 mV determines the value at which the concentration is half of the maximum value (Destexhe et al. 1994). The ionic AMPA and GABA _A currents are given by

$$ \begin{array}{llllllllll} &I_{\text{AMPA}}=g_{\text{AMPA}}N_{O}[V(t)-E_{\text{AMPA}}],\\ &I_{\text{GABA}_{A}}=g_{\text{GABA}_{A}}[N_{O_{1}}+N_{O_{2}}][V(t)-E_{\text{GABA}_{A}}], \end{array} $$

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where g_AMPA = g_GABA = 20 pS, E_AMPA = 0 mV, $E_{\text {GABA}_{A}}=-70$ mV, N _O and $N_{O_{1}}+N_{O_{2}}$ are the total number of open AMPA and GABA _A receptor channels, respectively (Destexhe et al. 1998b).