In this paper we show that a simple LRC oscillator, with inductive and resistive elements modulated by release of a relatively slow ionic species, can dynamically adapt its resonant frequency to that of a periodic stimulus. In combination with other, static, circuit elements, this plays a central role in producing a response that follows an omitted event by a fixed latency. An earlier study of linearized conductance-based models also found that subthreshold resonances can be tuned by conductance and capacitance parameters (Richardson et al., 2003
), although that oscillatory mechanism involves interactions between a single voltage equation and gating variables, while ours derives from membrane capacitance and inductance.
Our dual pathway model, including standard OFF bipolar cells and ON bipolar cells whose terminals adaptively resonate at the stimulus frequency, accounts for the omitted stimulus response (OSR) observed in isolated mouse and salamander retinas. The resonance arises from the interaction between voltage-gated calcium channels and calcium-gated potassium channels, as described in bipolar cells (Burrone and Lagnado, 1997
; Protti et al., 2000
) and other neurons (Art and Fettiplace, 1987
). Unlike previous models, the electrical resonance adapts its frequency using calcium-mediated feedback. Such a mechanism is expected to operate for any resonator involving BK-type potassium channels, which are gated both by voltage and calcium (Hille, 1992
; Jones et al., 1999
). Thus, the key novel feature relies on well-known properties of ion channels in bipolar cells (Burrone and Lagnado, 1997
; Sakaba et al., 1997
). Moreover, since the mechanism is basically linear, and does not rely on subtle current-voltage characteristics, it may be more generally responsible for predictive responses to disruptions in regular patterns such as those noted in the Introduction.
The model captures several qualitative features of the experimental observations. First, it produces OSRs with constant latencies from the expected flash for the frequency range 12 – 18 Hz (, upper panel). Second, over the lower frequency range, 8 – 13 Hz, latencies of model OSRs shift in the predictive direction, albeit not proportionally to the stimulus period (, lower panel), and OSR strengths are relatively smaller and less robust (robustness results not shown). Third, with fixed parameters the model can produce an OSR either for the high or low frequency range, but not both. All these behaviors match experimental data from individual ganglion cells. Finally, the spike-triggered averages and responses to flash sequences with the ON bipolar cells blocked match those observed for ganglion cells, and the OSR survives rectification in the OFF bipolar pathway prior to summation at the ganglion cell.
As described in Appendix E
, we also explored an alternative mechanism in which ganglion cells receive input from a spectrum of resonating bipolar cells with differing, but fixed, resonant frequencies. This resonator bank model generally fails to produce OSRs over a realistic frequency range. The addition of gap junction coupling can improve the range, but without producing constant latencies from expected flashes. This work does not rule out these or similar fixed parameter models, but it does suggest an important role for adaptation. It also shows that changes in stimuli can cause adaptation via calcium influx, with the natural prediction that manipulations that clamp intracellular calcium concentrations would lead to OSRs that cannot maintain a constant latency.
Our model includes desensitization in the OFF bipolar pathway, as has been observed experimentally to arise from the properties of ionotropic glutamate receptors (DeVries, 2000
). Because different degrees of desensitization are found for kainate- versus AMPA-type glutamate (DeVries and Schwartz, 1999
), we tried different degrees of desensitization in the model and found that the existence and timing of OSR was unaffected, even with no desensitization at all (results not shown). However, neglecting desensitization in the OFF pathway can lead to responses to second and succeeding flashes that are relatively large compared to the first flash response: behavior that is rarely observed. In addition, desensitization often caused the OFF bipolar pathway, alone, to fail to exceed threshold after the first flash (simulations not shown). This property is important to match the effects of the drug APB, which blocks ON bipolar cells, abolishing the OSR and
the sustained response (Schwartz and Berry II, 2008
In addition to bipolar cell terminals, adaptation can occur in photoreceptors, bipolar cell somas and ganglion cells. Our model ignores such detailed dynamics by describing the initial processing from photoreceptors to bipolar cell somas by a simple linear filter, and using a rectified linear relationship between terminal voltage and ganglion cell firing rate to produce the final output. Inclusion of these dynamics was not necessary to explain the salient features of the OSR, although they might be important to correctly predict firing rates of individual ganglion cells. The current model cannot produce OSRs for the more complex pattern violations observed in (Schwartz et al., 2007a
; Schwartz and Berry II, 2008
); nor does it produce the period doubling and higher harmonic patterns shown there. Addition of adaptive dynamics in other cells might solve this problem.
The adaptive resonator model was motivated specifically by the omitted stimulus response in the retina (Schwartz et al., 2007a
), but its feedback mechanism, in which calcium accumulation changes the effective inductance of calcium-gated potassium channels, may be quite generic. This mechanism relies on well-established properties of BK-type ion channels, which are found widely in the brain. Thus, the ability of a resonator to adapt its characteristic frequency over some range of input frequencies might be active in cellular oscillators in other subcortical and cortical areas (Llinas and Yarom, 1986
; Llinas, 1988
; Puil et al., 1994
; Gutfreud et al., 1995
; Hutcheon et al., 1996
; Pike et al., 2000
; Hutcheon et al., 1994
). An important factor is the accumulation of calcium with a time constant longer than each cycle of the stimulus, which allows the steady-state calcium level to track the stimulus frequency (). Cells with faster calcium dynamics will not significantly accumulate calcium, allowing resonators to be either fixed or adaptive depending on the calcium pumps and buffering.
Similar feedback may also be possible using the properties of Ih
channels (Robinson and Siegelbaum, 2003
). These channels are activated by hyperpolarization and produce a depolarizing current, which can drive a resonance. They also typically have long time constants, and so might encode stimulus frequency in a manner analogous to calcium accumulation in the current model. Ih
channels exhibit diverse expression patterns in bipolar cells, with mixtures of all four molecular types (HCN1-4) in different bipolar cells (Ivanova and Müller, 2006
), as well as in the axon terminals of ON bipolars (Fyk-Kolodziej and Pourcho, 2007
). Interestingly, Ih
channels can be gated by cyclic nucleotides in addition to voltage (Craven and Zagotta, 2006
). Because the metabotropic glutamate receptors on ON bipolar cells act via changes in cGMP levels, Ih
channels may interact in a complex way with this signaling cascade. Since our calcium feedback mechanism successfully captures the OSR observations, we have not attempted to add Ih
channels to our model.