Capacitance and temperature as bifurcation parameters for the saddle-

As mentioned before, the saddle-node-loop bifurcation can be induced by various system parameters. While a considerable part of the discussion considers the tempera-ture as bifurcation parameter (Chapter 10), the membrane capacitance, as an integral part of all conductance-based neuron models, is used as bifurcation parameter for the analysis in the second publication. The following sections establish the mathematical equivalence between membrane capacitance and a simplified temperature parameter, and set the latter into the broader context of temperature dependence in the nervous system.

3.5. Capacitance and temperature as bifurcation parameters for the saddle-node-loop bifurcation

3.5.1. Relative relaxation time constant as bifurcation parameter

The capacitance changes the velocities in the flow field of the dynamics in voltage dimension, while keeping the velocities in the gating dimensions unaffected. An increase in capacitance decreases the speed in voltage direction relative to the ion channel kinetics (Fig. 3.2). Slower dynamics in voltage direction effectively “squeezes”

the flow field along the voltage dimension as the velocity vector becomes steeper, leading to a limit cycle that is less broad in the voltage dimension (Fig. 3.2). As shown in detail in the second publication, the deformation of the flow field by the membrane capacitance will, under mild assumptions, induce an saddle-node-loop bifurcation.

The second paper and Sec. 8.1 show that the capacitance as such is already an interesting parameter that could be regulated by nature. Yet, instead of using the capacitance to slow down the voltage dynamics, one can equally well use a parameter to speed up the gating dynamics. A prime candidate for this is the “temperature parameter”φcommonly introduced in conductance-based models [38] as prefactor to the right side of the gating dynamics. For example, for a gating variableathis gives

da

dt =φa_∞−^a

τ_a . (3.4)

Whether the capacitance orφis used as bifurcation parameter, one will observe the same bifurcations, because the capacitance can be transformed into aφ-like parameter by a simple scaling in time, ˜t= t/C_m. With this new time scale, Eq. 3.1 becomes

dv

d˜t = I_input−^Iion(v,u), (3.5)

dm

d˜t =1/C_m1/τ_m(m_∞−^m), (3.6) dn

d˜t =1/Cm1/τn(n_∞−ⁿ), (3.7)

..., (3.8)

and with the identificationφ= 1/C_m, a set of equations as in Eq. 3.4 is recovered.

The parameterφis called “temperature parameter” because, as reviewed in the next section, a scaling of the gating time constants is one of the main effects of temperature (Sec. 3.5.2).

3.5.2. Temperature affects the relative time constant

Temperature affects virtually any biological process, because an increase in temperature generically increases chemical reaction rates and diffusion. Hence, for conductance-based models of voltage-gated ion channels, basically all parameters should show some form of temperature dependence. Yet, experimental results suggest that the qualitatively strongest effect of temperature is on the opening and closing rates of the ion channels⁷.

7Temperature influences not only ion channel kinetics and various other aspects of conductance-based ion channels, but also synaptic connections, neuronal properties and network dynamics, as reviewed

Observables such as gating rates tend to grow exponentially with an increase in temperature. A common measure for temperature dependence hence relies on the rela-tive change in the parameter when the temperature is increased by 10°C, the so called Q₁₀-values[155]. Opening and closing rates typically show Q₁₀values between two and four [70]. Moreover, temperature has a medium effect on the maximal conductances of leak and ion channels, and a negligible effect on the membrane capacitanceC_mand the reversal potentialsV_k[70]. On the level of single neurons, these and potentially also other effects of temperature influence the resting membrane potential, the membrane time constant, and the firing rate, as reviewed by Wang et al. [174]. Action potentials and post-synaptic excitatory potentials (EPSPs) are faster and change in amplitude.

For the conductance-based models used in Chapters 9 and 10, these various effects of temperature on neuronal dynamics are simplified, with a temperature dependency exclusively on the gating rates. To this end, the model definition Eq. 3.1 is augmented with the parameterφintroduced in Eq. 3.4. For simplicity, it is furthermore assumed that the scaling of the gating rates is equal for all gating variables, and that temperature does not affect the input. Note that a temperature dependence of the inputs would, for example, result from a network of connected, temperature-dependent neurons.

The assumption that the temperature dependence relies only on the gating rates (via the parameterφ) is quite restrictive. The following arguments show under which conditions this assumption can be relaxed. A rescaling of the time allows to take into account the temperature dependence of the maximal conductances and potentially the input current, and furthermore allows ion channel gating rates to differ between different gating variables, as reported experimentally. Temperature dependence is introduced in the general conductance-based model by scaling all conductancesg_iby φ_{g i} = Q^∆T/10_{g i} , inputI_extbyφ_ext = Q^∆T/10_ext and all right-hand sides of the gating vari-ables byφ_a =Q^∆T/10_a (whereQare different Q₁₀values for all parameters). Choosing φ_A = mina(φa)for some fixed temperature difference∆T, a rescaling of the time to

˜t=φ_Atreplaces Eq. 3.1 by

C_mdV d˜t = ¹

φ_A φ_extI_ext+φ_gleakg_leak(v−^Vleak) +

∑

k

φ_{g k}g_k a_k0...a_kj (v−^Vk)

! , da

d˜t = ^φa φ_A

a_∞−^a τ_a , ....

With this time scaling, it is obvious that, as long as max(φ_ext,φ_{g i})<min_a(φ_a) =φ_A, an increase in temperature will slow the voltage dynamics, and, for the gating variables, it will keep the rates constant or even increase them. Based on the results in the second publication, an approach of the saddle-node-loop bifurcation with temperature can hence be expected if max(Q_ext,Q_{g i})<min_a(Q_a).

by Wang et al. [174], modulating for example brain rhythms [129].