How Adaptation Makes Low Firing Rates Robust
- Arthur S. Sherman^{1}Email authorView ORCID ID profile and
- Joon Ha^{1}
https://doi.org/10.1186/s13408-017-0047-3
© The Author(s) 2017
Received: 1 December 2016
Accepted: 30 May 2017
Published: 24 June 2017
Abstract
Low frequency firing is modeled by Type 1 neurons with a SNIC, but, because of the vertical slope of the square-root-like f–I curve, low f only occurs over a narrow range of I. When an adaptive current is added, however, the f–I curve is linearized, and low f occurs robustly over a large I range. Ermentrout (Neural Comput. 10(7):1721-1729, 1998) showed that this feature of adaptation paradoxically arises from the SNIC that is responsible for the vertical slope. We show, using a simplified Hindmarsh–Rose neuron with negative feedback acting directly on the adaptation current, that whereas a SNIC contributes to linearization, in practice linearization over a large interval may require strong adaptation strength. We also find that a type 2 neuron with threshold generated by a Hopf bifurcation can also show linearization if adaptation strength is strong. Thus, a SNIC is not necessary. More fundamental than a SNIC is stretching the steep region near threshold, which stems from sufficiently strong adaptation, though a SNIC contributes if present. In a more realistic conductance-based model, Morris–Lecar, with negative feedback acting on the adaptation conductance, an additional assumption that the driving force of the adaptation current is independent of I is needed. If this holds, strong adaptive conductance is both necessary and sufficient for linearization of f–I curves of type 2 f–I curves.
Keywords
Adaptation SNIC bifurcation Firing rate1 Introduction
An alternative approach is to focus on adaptation currents that provide negative feedback to slow the firing rate. This respects the physiology of slow-firing neurons and is also needed to have actual adaption—the slowing of firing rate over time during a maintained stimulus. However, as shown in detail in [4], the presence of particular currents (they considered the A-type \(\mathrm{K}^{+}\) current) proposed by [5] is neither necessary nor sufficient to have low-frequency firing. They again emphasized the role of SNIC bifurcations.
The two approaches were married by theoretical analysis of HH-type models with various adaptation currents appended [6–9]. Our starting point is [9], where it was argued that the infinite slope of the f–I curve at the SNIC is, paradoxically, responsible for the ability of the adaptation current to reduce the firing rate and, moreover, accounts generically for the linear f–I curve of the adapted system.
We take another look here at spiking systems that have SNICs and are augmented with a slow adaptation variable, using phase-plane and bifurcation analysis. We aim in part to answer the question of why the lack of robustness is not merely transferred to the parameters of the adaptation variable and identify a geometric condition for avoiding this. We find that a SNIC in the system without adaptation does promote linearization, but that the f–I curve in the presence of adaptation may be linear only over a small interval unless the conductance of the adaptation current is sufficiently large. Finally we find that a SNIC is not necessary; type 2 systems in which oscillations arise from a Hopf bifurcation [2] can also show robust adaptation and linearization, though not generally low-frequency firing, if certain conditions hold.
2 Results
We consider first a very simple model with polynomial expressions instead of ionic currents, Hindmarsh–Rose (HR) [10], that has the essential components, a fast spiking subsystem and a slow adaptation variable. The adaptation equation is linear, which simplifies the application of averaging. We then extend the approach to a conductance-based HH-type model, Morris–Lecar (ML) [11], by reducing it to a form very similar to that of HR. This will show that linear adaptation, while convenient for the analysis, is not required for the effect.
2.1 Slow Firing in the Hindmarsh–Rose Model
Parameter values for SNIC and Hopf bifurcations with Hindmarsh–Rose model
SNIC | Hopf | |
---|---|---|
I | [0,20] | [−0.8,22] |
a | 1 | 1 |
b | 3.5 | 3.5 |
c | 1 | 1 |
d | 5.5 | 5.5 |
ϵ | 0.0005 | 0.00005 |
ϕ | 0.1 | 0.1 |
x̄ | −1.11 | −0.63 |
θ | 0 | 0.13 |
2.2 Approximating the Adapted Firing Rate Geometrically
We view this as a pseudo-phase plane for the full three-variable system and superimpose the steady-state (adapted) spiking solutions, which are accurately predicted by the intersection of the curve of average x, \(\langle x \rangle\) in Fig. 4.
To investigate this quantitatively, we fit straight lines to the steady-state f–I curves for several values of s over intervals of length 6 (Fig. 5C) or 10 and plotted the L_{2} error as a function of s for both long and short intervals (Fig. 5D). The error generally decreases with s, and larger s is needed for linearity over longer intervals. The non-monotonic behavior for small s is a consequence of the shapes of the curve of \(\langle x \rangle\) and the s nullcline (Fig. 5A). When s is small, the nullcline intersects the curve of \(\langle x \rangle\) on its linear portion, which helps to linearize the adapted f–I curve. (However, there is little reduction of firing rate, as the behavior of the f–I curve is dominated by the properties of the fast subsystem.) When s is large, the nullcline intersects the vertical portion of \(\langle x \rangle\), which again facilitates linearization. For intermediate values of s, however, the intersections occur along the nonlinear portion of \(\langle x \rangle \), especially for larger I, which inhibits linearization. These geometric relationships will play an important role later, when we address systems without a SNIC.
2.3 Approximating the Adapted Firing Rate by Averaging
Equation (4) incorporates the assumption that the firing rate depends only on the applied current as modified by the adaptive current, \(A(I)\); in addition we assume that A is an increasing function of I and that \(A(I) = 0\) at the threshold value of I, \(I_{*}\). The last is reasonable because the adaptation current responds to spike activity, and any baseline current could be absorbed into the other currents. Our goal is to predict the steady-state adapted firing rate but, in contrast to [8, 9], do not address the transient approach to the steady state.
2.3.1 Taylor Expansion Method
2.3.2 Mean Value Theorem Method
Claim 1
Given any interval \([I_{\star}, I^{\star}]\) on which \(\langle x \rangle\) is a monotonically decreasing function of z (cf. Fig. 5), \(A'(I) \rightarrow1\) uniformly as \(s \rightarrow\infty\)
If \(f_{0}^{\prime}(I_{\star}) = \infty\) (or is large because the unadapted system is near one with a SNIC), then \(f^{\prime}_{0}(\tilde{I})\) will also generally be large, and s will not need to be large to make \(A^{\prime}(I)\) near 1.
2.4 Adaptation When the Unadapted System Does Not Have a SNIC
2.5 Generalizing to a Conductance-Based Model
Parameter values for SNIC and Hopf bifurcations with Morris–Lecar model
SNIC | Hopf | |
---|---|---|
I | [40,100] | [58,80] |
\(g_{z}\) | 4 nS | 6 nS |
\(v_{k}\) | \(-84~\mbox{mV}\) | \(-84~\mbox{mV}\) |
\(v_{l}\) | \(-60~\mbox{mV}\) | \(-60~\mbox{mV}\) |
\(v_{ca}\) | \(120~\mbox{mV}\) | \(120~\mbox{mV}\) |
\(g_{k}\) | \(8~\mbox{nS}\) | \(8~\mbox{nS}\) |
\(g_{l}\) | \(2~\mbox{nS}\) | \(2~\mbox{nS}\) |
\(g_{ca}\) | \(4~\mbox{nS}\) | \(4~\mbox{nS}\) |
\(c_{m}\) | \(22~\mu\mbox{F}\) | \(22~\mu\mbox{F}\) |
\(v_{1}\) | \(-1.2~\mbox{mV}\) | \(-1.2~\mbox{mV}\) |
\(v_{2}\) | \(18~\mbox{mV}\) | \(18~\mbox{mV}\) |
\(v_{3}\) | \(12~\mbox{mV}\) | \(4~\mbox{mV}\) |
\(v_{4}\) | \(17~\mbox{mV}\) | \(20~\mbox{mV}\) |
s | 2 | 1.2 |
v̄ | −17 | 18 |
ε | 0.0001 | 0.0001 |
ϕ | 0.066667 | 0.066667 |
2.5.1 Morris–Lecar with SNIC
As in the HR case, the \(v_{\mathrm{equiv}}\) curve is vertical near the SNIC and for the same reasoning evoked for Eq. (13). Also as in HR, a linear f–I curve with substantial reduction in firing rate is obtained only when the w nullcline intersects the equivalent voltage curve along the vertical portion (blue curves), which only happens when \(g_{z}\) is sufficiently large. When this holds, the w nullcline picks off equally spaced values of w for equally spaced values of I. Assuming again that Δ is approximately constant, this implies that the adaptive current is linear in I, and this in turn yields the linear f–I curve. In fact, the adaptive current is linear for the large conductance but is nonlinear for the small conductance (not shown).
2.5.2 Morris–Lecar with HB
How can the results be so similar to the SNIC case when none of the assumptions needed to derive the properties of the SNIC case hold? One might think that it is because the system with HB is near one with a SNIC, but the time course of V near threshold is very different from the SNIC case—the spikes are distorted sinusoids with no long interspike interval (not shown).
An important clue is that, even though Δ is not constant in the unadapted (frozen) system for the HB case, Δ is very close to a constant in the adapted system for a large range of I near threshold, and that constant is \(\Delta(I_{\star})\). This can be seen from Fig. 12A, where the adapted f–I curve for large conductance samples values of f over \(I = [58, 80]\) that correspond to values of f achieved over \(I = [58, 63]\) in the unadapted system. The values of Δ are correspondingly limited to those attained in the unadapted system in the narrower range of I. In view of this, the near linearity of w does imply near linearity of the adaptive current. The near constancy of Δ is further demonstrated by the accurate prediction of the w locations of the trajectories in Fig. 12B, in which the w nullclines are plotted using \(\Delta(I_{\star})\).
The near constancy of Δ in the adapted system could have been predicted a priori because, independent of Claim 1, we should expect stretching of the f–I curve when \(g_{z}\) is large from the diagram in Fig. 6. We do not know whether the stretching is linear, but this observation justifies replacing \((v - E_{z})\) in Eq. (19) with \(\Delta(I_{\star})\) for the purpose of predicting the behavior of the full, adapted system. The argument of Claim 1, which does not depend on a SNIC or low frequency near threshold, then goes through, giving linear adaptive current and linear f–I for large \(g_{z}\).
A final point that requires explanation is why the \(v_{\mathrm{equiv}}\) curve has a nearly vertical portion near the threshold in the HB case. In the SNIC case, this follows from averaging (Eq. (13)) and, biophysically, from the lengthening of the interspike interval, which decreases \(\langle V \rangle\), as \(I \rightarrow I_{\star}\). This does not occur in the HB case, rather the sigmoidal shape we assumed for h comes into play. Mean v, and hence mean h (RHS of Eq. (22)), will tend to decrease as the SNIC is approached, but this increase may be gradual. However, if \(h(v) \approx0\) for v at the threshold, which is plausible, \(v_{\mathrm{equiv}}\) is forced to drop sharply to the flat region of h.
Note that in HR with HB (Fig. 9), where the activation of z is linear, \(x_{\mathrm{equiv}} = \langle x \rangle\), and the drop in \(x_{\mathrm{equiv}}\) is gradual. We have checked that if h is made linear in the ML system (Eq. (19)), \(v_{\mathrm{equiv}}\) is similarly gradual (not shown). The example of Fig. 9 shows that a vertical drop in \(v_{\mathrm{equiv}}\) is not necessary for a linear f–I curve. We will not attempt to account for all possible cases but conclude merely that a vertical drop in \(v_{\mathrm{equiv}}\) is not improbable in the HB case, and, if there is a vertical drop, f–I will be linearized and stretched when \(g_{z}\) is large enough.
3 Discussion
3.1 Context
Spike-frequency adaptation in neurons is well-studied in part because it is a basic and ubiquitous feature of neural behavior and in part because it contributes to information processing by networks of neurons. For example, in [6] it was shown to participate in forward masking, and in [14] local fatigue, which includes adaptation, was found to be responsible for switching between percepts in binocular rivalry. This in turn has generated interest in simplified models to facilitate simulation of large networks [8].
Others have focused on the ability of adaptation to linearize the f–I curve, because adapted neurons show this behavior and also because it has been found to have favorable properties in artificial neural networks for learning [15]. It was argued in [9] that linearization does not need to be imported into the system by assuming that the adaptation current was linear, as in [6]. Rather, linearization is a natural consequence of the square-root behavior of the unadapted f–I curve, which in turn comes from the presence of a SNIC in the unadapted spiking system.
As in previous analyses, we assume that adaptation is slow so that averaging can be applied, but we ask a different question: how does adaptation make low-frequency firing robust, that is, how is it maintained for a large range of input current? The main result that flowed from this question was that robustness and linearization both arise from adaptation because it stretches out the f–I curve. In retrospect this is natural because, as we learn in the first week of calculus, linearization is fundamentally a matter of stretching the scale of the independent variable. The role of stretching was previously illustrated in [8], their Fig. 8A, but was not made central to the theory.
Another way to linearize f–I curves that does not involve adaptation is noise, which can trigger firing at sub-threshold levels of I and smooth out a sharp threshold. See for example Fig. 1 in [16]. This is different in effect as well as mechanism from adaptation in that it achieves linearization by increasing firing at low I rather than reducing firing rate, so we will not address it further here.
3.2 Comparison to Previous Analyses
We confirmed the results in [9] that a SNIC in the unadapted system fosters linearization and robust reduction in firing rate. However, we showed numerically (Fig. 5D) that, whereas any degree of adaptation will result in linearization of the f–I curve, the size of the linear region depends continuously on the strength of adaptation (Fig. 5D). This showed that our concern about transferring parameter sensitivity to the adaptation equation was not unfounded and that it has a natural geometric interpretation. If the conductance is too low, then the nullcline of the adaptation variable in the Hindmarsh–Rose (HR) model will be nearly vertical (Fig. 5A), and the adapted system will be little different from the unadapted one (Fig. 5B). Similar but more complex graphs were made for the conductance-based Morris–Lecar (ML) model, in which the adaptation variable nullcline is nonlinear (Figs. 11B, 12B).
For the simple case of HR, in which the adaptation current has no fast voltage dependence (for example, no driving force), we showed further that a SNIC is not necessary if the adaptation conductance is large. If a SNIC is present, however, it would combine with the conductance to mediate linearization, so that the conductance need not be as large (Eq. (18)). The role of adaptation strength is intuitively obvious, and previously published numerical examples of linearization must have tacitly assumed it, but this feature was not revealed in previous analyses.
For ML with a SNIC, where voltage dependence comes into the adaptation current through the driving force (\(\Delta= v - E_{z}\) in Eq. (19)), we needed to assume that Δ is nearly constant. As argued in [9], this is likely to be a good approximation when there is a SNIC because v is nearly constant during the interspike interval, which dominates the oscillation period. If the unadapted system lacks a SNIC but is sufficiently near one that does have a SNIC, the firing rate would be low, and Δ should again be nearly constant. In [8] a more detailed analysis was carried out of this assumption and ways in which it may fail to hold, but we limit our consideration to cases where this is not a problem in order to focus on the essential features.
In addition to a SNIC, the analysis in [9] used the approximation that the adaptation current is proportional to the firing rate, as did [6] and [8]. This approximation was argued in [9] to follow from averaging the equation for the adaptation variable. In [8] it was shown that this may not always hold, depending on the voltage or calcium dependence of the adaptation current, and it was stated as a separate assumption (their Eq. (5.5)). This assumption is equivalent to the approximation we used in recasting the argument from [9], \(\langle x \rangle(z,I) = \beta f_{0}(I - z)\) (Eq. (13)), because z is proportional to \(\langle x \rangle\). (See Eq. (9); the offset x̄ is inconsequential as it could be absorbed into the applied current I and shifted to the x equation.) Note that we did not use this assumption in deriving the role of the adaptation conductance (Claim 1 and following text), but only the milder assumption that \(\langle x \rangle\) decreases monotonically with z.
The analysis of [9] essentially employed a linear approximation obtained by Taylor expansion around the SNIC. However, the Taylor approximation is only good when the adapted firing rate is nearly linear, and this is only assured when adaptation is strong. In [8], it was assumed tacitly that the slope of the unadapted f–I curve is sufficiently large in a sufficiently large neighborhood of threshold, as expected for a SNIC.
We circumvented this difficulty by applying the mean value theorem to the adaptation current, rather than approximating the firing rate itself. We showed that the adaptation current is nearly proportional to the applied current when the adaptation conductance is large (Claim 1). Our argument provided a uniform bound on the deviation from linearity as adaptation strength increases and also made the role of stretching more apparent (Eq. (17)). We did not have to make any assumption about the frequency dependence of the adaptation current.
Our analysis of stretching applies to HR in the type 2 (HB) case, but the strength of adaptation will generally have to be larger to achieve a linear f–I curve because there is no help from a SNIC (Fig. 10). Also, the firing rate defended by adaptation will not be 0, but whatever the threshold firing rate of the unadapted system happens to be.
However, our method does not apply to conductance-based, type 2 neurons because the voltage dependence of the adaptation current (present at least in the driving force) prevents use of the scaling argument we needed to transform the ML system to HR form. Nonetheless, adaptation and linearization can occur for sufficiently large adaptation conductance, as illustrated in Fig. 12. This happens because Δ is nearly constant for the adapted system even though it varies in the unadapted system. That in turn follows from the stretching, possibly nonlinear, of the f–I curve by the adaptive current (Fig. 6). Finally, this allows us to replace the type 2 system by one with constant driving force, and Claim 1 gives linear stretching as for type 1. The approximation breaks down for large enough I, but in practice it is good for a large range.
Our formulation for conductance-based models, with an adaptation current that is linear in a single gating variable, may not cover all possible cases, but it does include many typical ones, including two cases considered in [8] and [9], a voltage-dependent M-type \(\mathrm{K}^{+}\) current and an AHP current with a conductance that is linear in calcium. For adaptation currents with more complex voltage dependence, such as a slowly inactivating \(\mathrm {Na}^{+}\) current with fast gating variables, our theory may not apply even in the presence of a SNIC because the scaling argument used to derive Eq. (20) may not be valid even approximately.
3.3 Heuristic Summary
The larger \(g_{z}\) is, the closer a will be to 1, and the more strongly will the I axis be mapped toward 0, resulting in a linearized adaptation curve.
This result depends on \(v- E_{z}\) being nearly constant, which will hold if the unadapted system has a SNIC; together with large \(g_{z}\) this constitutes a sufficient set of conditions. If the unadapted system does not have a SNIC but has low-frequency firing near threshold, or an interspike interval that is much larger than the spike width, then large \(g_{z}\) is sufficient. Even if none of the above conditions apply, large \(g_{z}\) may be sufficient in many cases, as illustrated in Fig. 12. As observed in [8], there is unlikely to be a general theory to cover all type 2 systems.
3.4 Extensions
The SNIC plus slow negative feedback scenario is general, and should apply to many situations other than neuronal adaptation. One well-known candidate system is ER-driven calcium oscillations, which may exhibit frequency encoding of stimulus strength (ligand concentration) [17]. This can be achieved if the oscillation threshold is generated by a SNIC but is not robust. It has been suggested that control of oscillation frequency can be made more robust by adding a slow process to inhibit the IP3 receptor, specifically a calmodulin-dependent phosphorylation in Fig. 3B, [18]. Since calmodulin is activated by calcium, it would qualify as an activity-dependent adaptation process analogous to neuronal adaption, but this has not to our knowledge been modeled in detail.
Other means of achieving robustness have been considered theoretically. One is to average cell properties over a large population [19, 20]. That works well for a uniform but sloppy population of cells that need to synchronize to carry out a stereotypical task. For neuronal networks, in which individual cells may need to be constrained, the mechanism studied here, making parameters into variables, is more appropriate. A previous line of investigation had already introduced dynamic control of parameters, but differed in locating control at the level of gene expression [21]. Such regulation is slow, requiring tens of minutes to hours, whereas adaptation operates on the sub-second time scale.
What if a given adaptation process is not sufficiently strong? One solution is to increase the strength, but, if this is not feasible, an alternative is to make a parameter of the adaptation process itself into another slow variable. Chaining multiple negative feedback loops together should lead to a multiplicative improvement. This is also appropriate from the point of view of evolution, which cannot afford to rip out the knitting and start over. It is better to keep moving forward by adding new layers of control.
Declarations
Acknowledgements
The work was supported by the Intramural Research Program of the National Institutes of Health (NIDDK). We thank Bard Ermentrout and Jan Benda for helpful discussions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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