6 Adaptation and Firing Patterns

6.3 Biophysical Origin of Adaptation

We have introduced, in Section 6.1, formal adaptation variables wkw_{k} which evolve according to a linear differential equation (6.2). We now show that the variables wkw_{k} can be linked to the biophysics of ion channels and dendrites.

6.3.1 Subthreshold adaptation by a single slow channel

First we focus on one variable ww at a time and study its subthreshold coupling to the voltage. In other words, the aim is to give a biophysical interpretation of the parameters aa, τw\tau_{w}, and the variable ww that show up in the adaptation equation

τwdwdt=a(u-E0)-w.\tau_{w}\frac{{\text{d}}w}{{\text{d}}t}=a\,(u-E_{0})-w\,. (6.15)

The biophysical components of spike-triggered adaptation (i.e., the interpretation of the reset parameter bb) is deferred to Section 6.3.2. Here and in the following we write E0E_{0} instead of urestu_{\rm rest} in order to simplify notation and keep the treatment slightly more general.

As discussed in Chapter 2, neurons contain numerous ion channels (Section 2.3). Rapid activation of the sodium channels, important during the upswing of action potentials, is well approximated (Fig. 5.4) by the exponential nonlinearity in the voltage equation of the AdEx model, Eq. (6.3). We will see now that the subthreshold current ww is linked to the dynamics of other ion channels with a slower dynamics.

Let us focus on the model of a membrane with a leak current and a single, slow, ion channel, say a potassium channel of the Hodgkin-Huxley type

τmdudt=-(u-EL)-RLgKnp(u-EK)+RLIext\tau_{m}\frac{{\text{d}}u}{{\text{d}}t}=-(u-E_{L})-R_{L}\,g_{K}\,n^{p}\,(u-E_{% K})+R_{L}\,I_{\rm ext} (6.16)

where RLR_{L} and ELE_{L} are the resistance and reversal potential of the leak current, τm=RLC\tau_{m}=R_{L}C the membrane time constant, gKg_{K} the maximal conductance of the open channel and nn the gating variable (which appears with arbitrary power pp) with dynamics

dndt=-n-n0(u)τn(u).\frac{{\text{d}}n}{{\text{d}}t}=-\frac{n-n_{0}(u)}{\tau_{n}(u)}\,. (6.17)

As long as the membrane potential stays below threshold, we can linearize the equations (6.16) and (6.17) around the resting voltage E0E_{0}, given by the fixed point condition

E0=EL+(RLgK)n0(E0)pEK1+(RLgK)n0(E0)p.E_{0}={E_{L}+(R_{L}\,g_{K})\,n_{0}(E_{0})^{p}\,E_{K}\over 1+(R_{L}\,g_{K})\,n_% {0}(E_{0})^{p}\,}\,. (6.18)

The resting potential is shifted with respect to the leak reversal potential if the channel is partially open at rest, n0(E0)>0n_{0}(E_{0})>0. We introduce a parameter β=gKpn0(E0)p-1(E0-EK)\beta=g_{K}\,p\,n_{0}(E_{0})^{p-1}\,(E_{0}-E_{K}) and expand n0(u)=n0(E0)+n0(u-E0)n_{0}(u)=n_{0}(E_{0})+n_{0}^{\prime}\,(u-E_{0}) where n0n_{0}^{\prime} is the derivative dn0/dudn_{0}/du evaluated at E0E_{0}.

The variable w=β[n-n0(E0)]w=\beta\,[n-n_{0}(E_{0})] then follows the linear equation

τn(E0)dwdt=a(u-E0)-w.\tau_{n}(E_{0})\frac{{\text{d}}w}{{\text{d}}t}=a\,(u-E_{0})-w\,. (6.19)

We emphasize that the time constant of the variable ww is given by the time constant of the channel at the resting potential. The parameter aa is proportional to the sensitivity of the channel to a change in the membrane voltage, as measured by the slope dn0/dudn_{0}/du at the equilibrium potential E0E_{0}.

The adaptation variable ww is coupled into the voltage equation in the standard form

τmeffdudt=-(u-E0)-Rw+RIext.\tau_{m}^{\rm eff}\frac{{\text{d}}u}{{\text{d}}t}=-(u-E_{0})-R\,w+R\,I_{\rm ext}. (6.20)

Note that the membrane time constant and the resistance are rescaled by a factor [1+(RLgK)n0(E0)p]-1[1+(R_{L}\,g_{K})\,n_{0}(E_{0})^{p}\,]^{-1} with respect to their values in the passive membrane equation, Eq. ((6.16)). In fact, both are smaller because of partial opening of the channel at rest.

In summary, each channel with nonzero slope dn0/dudn_{0}/du at the equilibrium potential E0E_{0} gives rise to an effective adaptation variable ww. Since there are many channels, we can expect many variables wkw_{k}. Those with similar time constants can be summed and grouped into a single equation. But if time constants are different by an order of magnitude or more then several adaptation variables are needed, which leads to the model equations (6.1) and (6.2).

6.3.2 Spike-triggered adaptation arising from a biophysical ion-channel

Type Fig. act./inact. τw\tau_{w} (ms) β\beta (pA) a (nS) δx\delta_{x} bb (pA)
INaI_{\rm Na} 2.3 inact. 20 -120 5.0 - -
IMI_{\rm M} 2.13 act. 61 12 0.0 0.0085 0.1
IAI_{\rm A} 2.14 act. 33 12 0.3 0.04 0.5
IHVA+IK[Ca]I_{\rm HVA}+I_{\rm K[Ca]} 2.16 act. 150 12 0 0.05 0.6
IhI_{h} 2.18 inact. 8.5 -48 0.8 - -
INaSI_{\rm NaS} 2.19 act 200 -120 -0.08 0.0041 -0.48
Table 6.2: Parameter values for ion channels presented in Chapter 2 for model linearized around -65-65 mV for Rgk=1Rg_{k}=1. The action potential is assumed to consist of a pulse of 11 ms duration at 00 mV. The approximation to obtain δx\delta_{x} and bb is valid only when τx(0\tau_{x}(0mV)) is significantly larger than one millisecond.

We have seen in Chapter 2 that some ion channels are partially open at the resting potential, while others react only when the membrane potential is well above the firing threshold. We now focus on the second group in order to give a biophysical interpretation of the jump amplitude bb of a spike-triggered adaptation current.

Let us return to the example of a single ion channel of the Hodgkin and Huxley type such as the potassium current in Eq. (6.16). In contrast to the treatment before, we now study the change in the state of the ion channel induced during the large-amplitude excursion of the voltage trajectory during a spike. During the spike, the target n0(u)n_{0}(u) of the gating variable is close to one; but since the time constant τn\tau_{n} is long, the target is not reached during the short time that the voltage stays above the activation threshold. Nevertheless, the ion channel is partially activated by the spike. Unless the neuron is firing at a very large firing rate, each additional spike activate the channel further, always by the same amount Δn\Delta_{n}, which depends on the duration of the spike and the activation threshold of the current (Table 6.2). The spike-triggered jump in the adapting current ww is then

b=βΔn.b=\beta\Delta_{n}\,. (6.21)

where β=gKpn0(E0)p-1(E0-EK)\beta=g_{K}\,p\,n_{0}(E_{0})^{p-1}\,(E_{0}-E_{K}) has been defined before.

Fig. 6.10: Another type of bursting in a model with two spike-triggered currents. A Voltage trace of the neuron model Eq. (6.3)-(6.4) with ureset=-55u_{\rm reset}=-55 mV, ϑrh=-50\vartheta_{\rm rh}=-50 mV, b1=-12b_{1}=-12 pA, b2=60b_{2}=60 pA, τ1=20\tau_{1}=20 ms, τ2=61\tau_{2}=61 ms, a1=-3a_{1}=-3 nS and a2=0a_{2}=0. Parameters were chosen to correspond to a neuron coupled with a dendritic compartment and IMI_{\rm M}. B. Voltage deflection brought by an isolated spike. Each spike brings first refractoriness, then a facilitation and finally adaptation on a longer timescale.

Again, real neurons with their large quantity of ion channels have many adaptation currents wkw_{k}, each with its own time constant τk\tau_{k}, subthreshold coupling aka_{k} and spike-triggered jump bkb_{k}. The effective parameter values depend on the properties of the ion channels (Table 6.2).

Example: Calculating the jump bb of the spike-triggered adaptation current

We consider a gating dynamics

   d   n   d   t=-n-n0(u)τn(u).\frac{{\text{d}}n}{{\text{d}}t}=-\frac{n-n_{0}(u)}{\tau_{n}(u)}\,. (6.22)

with the steplike activation function n0(u)=Θ(u-u0act)n_{0}(u)=\Theta(u-u_{0}^{\rm act}) where u0act=-30u_{0}^{\rm act}=-30mV and τn(u)=100\tau_{n}(u)=100ms independent of uu. Thus, the gating variable nn approaches a target value of 1 whenever the voltage uu is above the activation threshold u0actu_{0}^{\rm act}. Since the activation threshold of -30mV is above the firing threshold (typically in the range of -40mV) we can safely state that the neuron activation of the channel can only occur during an action potential. Assuming that, during an action potential the voltage remains for Δt=1\Delta_{t}=1ms above u0actu_{0}^{\rm act}, we can integrate Eq. (6.22) and find that each spike causes an increase Δn=Δt/τn\Delta_{n}=\Delta_{t}/\tau_{n} where we have exploited that Δtτn\Delta_{t}\ll\tau_{n}. If we plug in the above numbers, we see that each spike causes an increase of nn by a value of 0.01. If the duration of the spike were twice as long, the increase would be 0.02. After the spike the gating variable decays with the time constant τn\tau_{n} back to zero. The increase Δn\Delta_{n} leads to a jump amplitude of the adaptation current given by Eq. (6.21).

6.3.3 Subthreshold adaptation caused by passive dendrites

While in the previous paragraph, we have focused on the role of ion channels, here we show that a passive dendrite can also give rise to a subthreshold coupling of the form of Eq. (6.15).
We focus on a simple neuron model with two compartments, representing the soma and the dendrite, superscripts ss and dd respectively. The two compartments are both passive with membrane potential Vs, VdV^{s},\,V^{d}, transversal resistance RTs, RTdR_{\text{T}}^{s},\,R_{\text{T}}^{d}, capacity Cs, CdC^{s},\,C^{d} and resting potential ur, Edu_{\rm r},\,E^{d}. The two compartments are linked by a longitudinal resistance RLR_{\text{L}} (see Chapter 3). If current is injected only in the soma, then the two-compartment model with passive dendrites corresponds to

ddtVs\displaystyle{{\text{d}}\over{\text{d}}t}V^{s} =\displaystyle= 1Cs[-(Vs-urest)RTs-Vs-VdRL+I(t)],\displaystyle\frac{1}{C^{s}}\,\left[-{(V^{s}-u_{\rm rest})\over R_{\text{T}}^{% s}}-{V^{s}-V^{d}\over R_{\text{L}}}+I(t)\right]\,, (6.23)
ddtVd\displaystyle{{\text{d}}\over{\text{d}}t}V^{d} =\displaystyle= 1Cd[-(Vd-Ed)RTd-Vd-VsRL].\displaystyle\frac{1}{C^{d}}\,\left[-{(V^{d}-E^{d})\over R_{\text{T}}^{d}}-{V^% {d}-V^{s}\over R_{\text{L}}}\right]\,. (6.24)

Such a system of differential equations can be mapped to the form of Eq. (6.15) by considering that the variable ww represents the current flowing from the dendrite into the soma. In order to keep the treatment transparent, we assume that Ed=urest=EE^{d}=u_{\rm rest}=E. In this case the adaptation current is w=-(Vd-urest)/RLw=-(V^{d}-u_{\rm rest})/R_{\text{L}} and the two equations above reduce to

τeffdVsdt\displaystyle\tau^{\rm eff}{dV^{s}\over dt} =\displaystyle= -(Vs-E)-Reffw\displaystyle-(V^{s}-E)-R^{\rm eff}\,w (6.25)
τwdwdt\displaystyle\tau_{w}{dw\over dt} =\displaystyle= a(Vs-E)-w\displaystyle a\,(V^{s}-E)-w (6.26)

with an effective input resistance Reff=RTs/[1+(RTs/RL)]R^{\rm eff}=R^{s}_{\rm T}/[1+(R^{s}_{\rm T}/R_{\rm L})], an effective somatic time constant τeff=CsReff\tau^{\rm eff}=C^{s}\,R^{\rm eff} an effective adaptation time constant τw=RLCd/[1+(RL/RD)]\tau_{w}=R_{\rm L}\,C^{d}/[1+(R_{\rm L}/R_{\rm D})] and a coupling between somatic voltage and adaptation current a=-[RL+(RL2/RD)]-1a=-[R_{\rm L}+(R^{2}_{\rm L}/R_{\rm D})]^{-1}.

There are three conclusions we should retain from this mapping. First, aa is always negative, which means that passive dendrites introduce a facilitating subthreshold coupling. Second, facilitation is particularly strong with a small longitudinal resistance. Third, the timescale of the facilitation τw\tau_{w} is smaller than the dendritic time constant RTdCdR^{d}_{\text{T}}C^{d} - so that, compared to other ‘adaptation’ currents, the dendritic current is a relatively fast one.

In addition to the subthreshold coupling discussed here, dendritic coupling can also lead to a spike-triggered current as we will see in the next example.

Example: Bursting with a Passive Dendrite and IMI_{\rm M}.

Suppose that the action potential can be approximated by a one-millisecond pulse at 0 mV. Then each spike brings an increase in the dendritic membrane potential. In terms of the current ww, the increase is b=-aE0(1-e1 ms/τw)b=-aE_{0}(1-e^{1\text{ ms}/\tau_{w}}). Again, the spike-triggered jump is always negative, leading to spike-triggered facilitation. Figure 6.10 shows an example where we combined a dendritic compartment with the linearized effects of the M-current (Table 6.2) to result in regular bursting. The bursting is mediated by the dendritic facilitation which is counterbalanced by the adapting effects of IMI_{\rm M}. The firing pattern looks different to the bursting in the AdEx (Fig. 6.4) as there is no alternation between detour and direct resets. Indeed, many different types of bursting are possible (see (238)). This example (especially Fig. 6.10B) suggests that the dynamics of spike-triggered currents on multiple timescales can be understood in terms of their stereotypical effect on the membrane potential - and this insight is the starting point for the Spike Response Model in the next section.