Encoding the fine-structured mechanism of action potential dynamics with qualitative motifs

Abstract

This work presents a neuroinformatic method for deriving mechanistic descriptions of fine-structured neural activity. This is a new development in the computer-assisted analysis of dynamics in conductance-based models, which is illustrated using single compartment models of an action potential. A sequence of abstract, qualitative motifs is inferred from this analysis, forming a template that is independent of the specific equations from which they were abstracted. The template encodes the assumptions behind the model reduction steps used to derive the motifs, and so specifies quantitative information about their domains of validity. The template representation of a mechanism is converted to a hybrid dynamical system, which is simulated as a sequence of low-dimensional reduced models (in this example, phase plane models) with appropriate switching conditions taken from the motifs. We demonstrate the validity of the template on a detailed single neuron model of spiking taken from the literature, and show that the corresponding hybrid system simulation closely mimics the spiking dynamics of the full model.

Appendices

Appendix A: Model kinetics and parameters

For all three test models, the total ionic current is $\sum I_{\textrm{\tiny ionic}} = g_{Na} m h^3 (V-E_{Na}) + g_K n^4 (V-E_K) + g_L (V - E_L)$. For the Wang-Buzsáki model, m = m _∞(V), otherwise all gating variables are given by Eq. (2). The channel rate kinetics of any gating variable s are converted into s _∞ and τ _s according to the standard definitions, s _∞ = α _s / (α _s  + β _s ) and τ _s  = 1 / (α _s  + β _s ), for the forward and backward rate functions α _s and β _s , respectively.

1.1 A.1 Classic Hodgkin-Huxley model

$$\begin{array}{rll} \alpha_m & = & 0.1(V+40)/(1-\exp(-(V+40)/10)) \\ \beta_m & = & 4\exp(-(V+65)/18) \\ \alpha_h & = & 0.07\exp(-(V+65)/20) \\ \beta_h & = & 1/(1+\exp(-(V+35)/10)) \\ \alpha_n & = & 0.01(V+55)/(1-\exp(-(V+55)/10)) \\ \beta_n & = & 0.125\exp(-(V+65)/80) \end{array}$$

Parameter values: g _Na  = 120, g _K  = 36, g _L  = 0.3, E _Na  = 50, E _K  = − 77, E _L  = − 54.4, $I_{\textrm{\tiny applied}} = 8$, C = 1.

1.2 A.2 Wang-Buzsáki interneuron model

$$\begin{array}{rll} \alpha_m & = & 0.1(V+35)/(1-\exp(-(V+35)/10)) \\ \beta_m & = & 4\exp(-(V+60)/18) \\ \alpha_h & = & 0.35\exp(-(V+58)/20) \\ \beta_h & = & 5/(1+\exp(-(V+28)/10)) \\ \alpha_n & = & 0.05(V+34)/(1-\exp(-(V+34)/10)) \\ \beta_n & = & 0.625 \exp(-(V+44)/80) \end{array}$$

Parameter values: g _Na  = 35, g _K  = 9, g _L  = 0.1, E _Na  = 55, E _K  = − 90, $E_L = -65, I_{\textrm{\tiny applied}}=2.5, C = 1$.

1.3 A.3 Alternative interneuron model

$$\begin{array}{rll} \alpha_m & = & 0.32(V+54)/(1-\exp(-(V+54)/4)) \\ \beta_m & = & 0.28(V+27)/(\exp((V+27)/5)-1) \\ \alpha_h & = & 0.128\exp(-(50+V)/18) \\ \beta_h & = & 4/(1+\exp(-(V+27)/5)) \\ \alpha_n & = & 0.032(V+52)/(1-\exp(-(V+52)/5)) \\ \beta_n & = & 0.5\exp(-(57+V)/40) \end{array}$$

Parameter values: g _Na  = 100, g _K  = 80, g _L  = 0.1, E _Na  = 100, E _K  = − 100, E _L  = − 67, $I_{\textrm{\tiny applied}}=2$, C = 1.

1.4 A.4 Leech heart interneuron

The details of the channel kinetics are given in Cymbalyuk et al. (2002). All channels are given by the standard Hodgkin-Huxley formalism. The parameters for the tonic spiking regime of the HN interneuron model were: g _L  = 8.0, E _L  = − 60, g _CaF  = 5, g _CaS  = 3.2, E _Ca  = 135, g _K1 = 100, g _K2 = 80, g _KA  = 80, E _K  = − 70, g _H  = 4, E _H  = − 21, g _P  = 7, E _Na  = 45, g _Na  = 200, $I_{\textrm{\tiny applied}}=0$, C = 0.5.

Appendix B: Dominant scale sets for the original neuron models

The ${\mathcal{A}_\Psi}$ and ${\mathcal{A}_\Omega}$ sets for the periodic orbit of the Type II Hodgkin-Huxley model are shown in Tables 2 and 3, using σ = γ = 4. The time intervals for each epoch are not contiguous due to the discrete time points from the numerically computed periodic orbit. However, the dominant scales method does not require the transitions to be computed more accurately. A graphic view of these transitions overlaid on the periodic orbit is presented in Fig. 2. Corresponding sets for the Type I HH and Wang-Buzsáki models are shown in Tables 4–7. In these tables, variables are labeled fast or slow only if they belong to ${\mathcal{F}}$ or ${\mathcal{S}}$ for the entire epoch. All times are shown to the nearest 1/100 ms.

Table 2 Ordered sets of active and modulatory variables during one period of the Type II HH model orbit, using Ψ

Table 3 Ordered sets of active and modulatory variables during one period of the Type II HH model orbit, using Ω

Table 4 Ordered sets of active and modulatory variables during one period of the Type I HH model orbit, using Ψ

Table 5 Ordered sets of active and modulatory variables during one period of the Type I HH model orbit, using Ω

Table 6 Ordered sets of active and modulatory variables during one period of the Type I WB model orbit, using Ψ

Table 7 Ordered sets of active and modulatory variables during one period of the Type I WB model orbit, using Ω