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• W4239572886 abstract "In this issue of Neuron, Remme and colleagues examine the biophysics of synchronization between oscillating dendrites and soma. Their findings suggest that oscillators will quickly phase-lock when weakly coupled. These findings are at odds with assumptions of an influential model of grid cell response generation and have implications for grid cell response mechanisms. In this issue of Neuron, Remme and colleagues examine the biophysics of synchronization between oscillating dendrites and soma. Their findings suggest that oscillators will quickly phase-lock when weakly coupled. These findings are at odds with assumptions of an influential model of grid cell response generation and have implications for grid cell response mechanisms. As our moon orbits the Earth, it rotates. Yet on Earth we see only one face of the moon. This happens because the moon happens to rotate by exactly the same amount that it revolves. The matching of angular speeds for rotation and revolution is no coincidence. It is the inexorable result of the periodic movements of the earth and moon combined with the weak gravitational tidal forces coupling them. In the language of the theory of coupled oscillators, the moon's rotation and revolution have converged to the stable phase-locked solution. In this issue of Neuron, Remme et al., 2010Remme M.W.H. Lengyel M. Gutkin B.S. Neuron. 2010; 66 (this issue): 429-437Abstract Full Text Full Text PDF PubMed Scopus (49) Google Scholar use the theory of weakly coupled oscillators to provide a compelling analysis of the biophysical viability of an influential model of grid cell response generation. Rats and mice (Fyhn et al., 2008Fyhn M. Hafting T. Witter M.P. Moser E.I. Moser M.-B. Hippocampus. 2008; 18: 1230-1238Crossref PubMed Scopus (105) Google Scholar and references therein) have grid cells, and there is good evidence for their presence in humans (Doeller et al., 2010Doeller C.F. Barry C. Burgess N. Nature. 2010; 463: 657-661Crossref PubMed Scopus (457) Google Scholar). A single grid cell responds as a function of animal location in two-dimensional (2D) space, with a firing peak at every vertex of a (virtual) regular triangular lattice that covers the plane. The spatial period of the grid cell response is independent of animal speed. Models of grid cell activity fall into two main classes, both predicated on the hypothesis that position-coded grid cell responses are obtained using animal-velocity cues. Aside from this shared hypothesis, the model classes are disparate in their assumptions and predictions, with each class explaining largely complementary subsets of grid cell properties. One model class assumes that strong network-level recurrent connectivity unleashes a spontaneous patterning of the neural population response (Fuhs and Touretzky, 2006Fuhs M.C. Touretzky D.S. J. Neurosci. 2006; 26: 4266-4276Crossref PubMed Scopus (431) Google Scholar, McNaughton et al., 2006McNaughton B.L. Battaglia F.P. Jensen O. Moser E.I. Moser M.-B. Nat. Rev. Neurosci. 2006; 7: 663-678Crossref PubMed Scopus (1285) Google Scholar, Burak and Fiete, 2009Burak Y. Fiete I.R. PLoS Comput. Biol. 2009; 5: e1000291Crossref PubMed Scopus (414) Google Scholar, and references therein). These population responses translate into spatially periodic responses of single neurons. The other model class assumes that interfering temporal oscillations set up a beat wave that can be mapped onto space to produce spatially periodic grid responses (Burgess et al., 2007Burgess N. Barry C. O'Keefe J. Hippocampus. 2007; 17: 801-812Crossref PubMed Scopus (496) Google Scholar, Hasselmo, 2008Hasselmo M.E. Hippocampus. 2008; 18: 1213-1229Crossref PubMed Scopus (156) Google Scholar). Remme and colleagues analyze an exemplar of the temporal interference (TI) models, based on voltage oscillations within a single neuron (Burgess et al., 2007Burgess N. Barry C. O'Keefe J. Hippocampus. 2007; 17: 801-812Crossref PubMed Scopus (496) Google Scholar). The model may be summarized as follows: if the soma oscillates at a fixed temporal frequency, and a dendrite oscillates at a slightly different frequency that increases linearly with the running speed of the animal, then the summed response of the soma and dendrite, when plotted as a function of animal location as the animal runs in a straight line (1D), is periodic and invariant to running speed (Figure 1, top right). The generalization to 2D and to a triangular lattice pattern in space comes from assuming three independently oscillating dendritic branches, modulated in frequency by the component of the animal's velocity along multiples of 120°, respectively. The output of the three branches, when summed, produces a regular triangular lattice pattern characteristic of grid cells. The summation is assumed to occur at the soma, and the dendrites are assumed to be independent intrinsic oscillators whose frequency is determined solely by the external velocity input. The latter assumption is the focus of the study by Remme and colleagues. In addition to reproducing the spatial aspects of a grid cell's response, the TI models generate for “free” (as an essential consequence) the phenomenon of phase precession, seen in grid cells of entorhinal layer II (but not layer III) (Fyhn et al., 2008Fyhn M. Hafting T. Witter M.P. Moser E.I. Moser M.-B. Hippocampus. 2008; 18: 1230-1238Crossref PubMed Scopus (105) Google Scholar). TI models additionally predict that grid cells with a larger spatial period must display lower temporal oscillation frequencies. Experiments in the dorsolateral band of the entorhinal cortex showed that the intrinsic temporal response frequency of neurons increases systematically toward the ventral end (Giocomo and Hasselmo, 2008Giocomo L.M. Hasselmo M.E. J. Neurosci. 2008; 28: 9414-9425Crossref PubMed Scopus (88) Google Scholar, Jeewajee et al., 2008Jeewajee A. Barry C. O'Keefe J. Burgess N. Hippocampus. 2008; 18: 1175-1185Crossref PubMed Scopus (142) Google Scholar), in concert with the observed increase in spatial period of grid cells and with the prediction of the subthreshold voltage version of the TI models. For these reasons, the subthreshold TI model has fueled experimental study and initiated a nascent but growing understanding of the cellular determinants of grid cell response. However, Remme and colleagues show that the subthreshold TI model's assumptions cannot easily be reconciled with neural biophysics. When two oscillators with similar intrinsic frequencies are weakly coupled, they will become phase-locked, as observed by Christiann Huygens for two pendulum clocks hung from the same wooden beam (Figure 1, bottom left) and noted in his 1665 correspondence to the Royal Society. The universality of the dynamics of weakly coupled oscillators predicts phase-locking (Pikovsky et al., 2002Pikovsky A. Rosenblum M. Kurths J. Am. J. Phys. 2002; 70 (655–655)Crossref Google Scholar), whether the oscillators in question are gravitational bodies, pendula, chemical reactions, or as shown previously by Remme and colleagues, distant dendritic compartments interacting through the cell membrane (Remme et al., 2009Remme M.W.H. Lengyel M. Gutkin B.S. PLoS Comput. Biol. 2009; 5: e1000493Crossref PubMed Scopus (43) Google Scholar). (If the coupling between oscillators is strong, other interesting phenomena such as “oscillator death” and chaos can occur, but these effects are not considered in the present work because the dendritic interactions are estimated to be in the weak coupling regime.) Therefore, the first conclusion of Remme and colleagues is that is not possible for a grid cell to indefinitely maintain the frequency and phase differences between dendritic compartments required for the perpetual generation of grid responses. The ultimate stability of only the phase-locked solution need not exclude the possibility that dendrites may transiently remain independent, for long enough to retain their position-coded phases and produce grid patterns over the 10–20 min during which grid patterns are recorded. The more weakly coupled a pair of oscillators, the more slowly the phase-locked solution is reached. For instance, the same tidal forces that drove the moon's rotation and revolution into phase-locked states are also slowing the Earth's rotation until it phase-locks with the moon. Eventually, the moon will also only see one face of the Earth. Nevertheless, because of the moon's relatively small gravitational effect on the earth, the Earth is slow to phase-lock and still rotates out of step with the moon. Addressing the question of timescale for phase-locking between dendritic compartments is the central focus of the present work by Remme and colleagues. They use both theoretical and numerical analyses to arrive at quantitative, well-parameterized results that can be grounded in experimental data on membrane properties, synaptic currents, and cell morphology. The basic setup of their analytical model is a pair of dendritic compartments modeled as intrinsic oscillators, coupled by a cable equation. Each dendritic oscillator's phase resetting curve (PRC) defines the change in the phase of the oscillator induced by an infinitesimal input, as a function of the oscillator's present phase. Inputs to the dendrite affect its phase through its PRC. In this model, synaptic velocity inputs change the phase of the ongoing dendritic oscillation, and thus can also alter its instantaneous frequency. Similarly, the state of the other dendrite also affects dendritic phase through the voltage of the cable that couples them. In a theoretical analysis of the transient dynamics of this coupled oscillator system, Remme and colleagues identify a fundamental trade-off between “independence” and “democracy” in dendritic computation (Figure 1, bottom right). If the dendrites strongly influence somatic voltage (democracy), as required for generating the beat wave in the TI model, then by the bidirectionality of electrical coupling, the soma will also affect the dendritic oscillation (loss of independence), degrading the velocity-defined phase information about position contained in the dendrite. In other words, altering dendritic properties to increase independence comes at the cost of a decrease in the influence of (1) the dendrite on the soma (which translates into a decrease of the amplitude of the beat wave) and of (2) synaptic inputs onto the dendrite. Remme et al. next numerically simulate the voltage dynamics in a morphologically correct spiny stellate cell with active and passive conductances. They show that different parameter combinations sample different points on a set of independence-democracy trade-off curves (Figure 1, bottom right). None of the parameters in their model, varied over biophysically reasonable values, enables the coexistence of sufficiently high levels of democracy and independence to generate spatially periodic grid responses. Is there a way to reconcile the conflict between the biophysics of neural oscillators and the requirements of the subthrehsold TI model? If the oscillations in the dendritic compartment were produced by oscillatory synaptic input, which unidirectionally drives neural voltage without being affected in return by postsynaptic voltage, it might be possible to realize a regime of strong dendritic influence on the soma and independent dendritic phase. Such alterations call for network mechanisms or the existence of separate cells with temporally periodic outputs whose frequency or phases are modulated by animal speed. Other TI models, including one based on the interference of regular spike trains from persistent spiking neurons of the entorhinal cortex (Hasselmo, 2008Hasselmo M.E. Hippocampus. 2008; 18: 1213-1229Crossref PubMed Scopus (156) Google Scholar), are built on some of these alternative assumptions and generate similar predictions. Unfortunately, all forms of TI models are subject to another biophysical feasibility issue (raised in Welinder et al., 2008Welinder P.E. Burak Y. Fiete I.R. Hippocampus. 2008; 18: 1283-1300Crossref PubMed Scopus (64) Google Scholar and explored in Zilli et al., 2009Zilli E.A. Yoshida M. Tahvildari B. Giocomo L.M. Hasselmo M.E. PLoS Comput. Biol. 2009; 5: e1000573Crossref PubMed Scopus (43) Google Scholar), again related to the assumption that spatial information can be faithfully represented in the phase of a temporal oscillation over extended periods of time. Single biological oscillators are typically noisy. The theta peak in the local field potential spectrum is broad, signifying a variable oscillation period and phase loss. Membrane potential oscillations in single entorhinal cells are similarly variable: over fewer than 10 cycles, information about the initial phase is lost (analyzed in Welinder et al., 2008Welinder P.E. Burak Y. Fiete I.R. Hippocampus. 2008; 18: 1283-1300Crossref PubMed Scopus (64) Google Scholar). Even persistent spiking neurons of the entorhinal cortex, despite their low interspike interval variance, would experience loss of phase information at a rate too high for generating grid responses without assuming network-level contributions to grid activity beyond the interference of spikes from cell triplets (Zilli et al., 2009Zilli E.A. Yoshida M. Tahvildari B. Giocomo L.M. Hasselmo M.E. PLoS Comput. Biol. 2009; 5: e1000573Crossref PubMed Scopus (43) Google Scholar). At this point, we may be left wondering, “What are we to conclude about the mechanisms underlying grid cell responses? And with these biophysical caveats on TI models, why are their predictions so successful?” Like the TI models, recurrent neural network models of grid cell response based on continuous attractor dynamics use velocity inputs to produce periodic grid-like responses. In addition, they necessarily generate groups of cells with identical periods and predict that cells with the same spatial period must have identical orientations but all possible spatial phases, as found in experiments. Cells with combined velocity and grid tuning are another natural byproduct and prediction of the network models (Fuhs and Touretzky, 2006Fuhs M.C. Touretzky D.S. J. Neurosci. 2006; 26: 4266-4276Crossref PubMed Scopus (431) Google Scholar), and their existence was verified by experiment (Sargolini et al., 2006Sargolini F. Fyhn M. Hafting T. McNaughton B.L. Witter M.P. Moser M.-B. Moser E.I. Science. 2006; 312: 758-762Crossref PubMed Scopus (928) Google Scholar). The models also have some robustness to neural noise in the computation of position-coded phase from velocity over minutes (Burak and Fiete, 2009Burak Y. Fiete I.R. PLoS Comput. Biol. 2009; 5: e1000291Crossref PubMed Scopus (414) Google Scholar). For these reasons, recurrent network dynamics remain a viable model for the generation of spatially periodic grid responses. However, present recurrent network models do not include mechanisms for phase precession. At the same time, over the short durations of a few theta cycles on which phase precession occurs, the biophysical forces that ultimately lead to phase-locking have relatively little effect on soma-dendritic dynamics and are unimportant. Combining these observations into a composite model in which temporal oscillations and recurrent dynamics play separable roles in explaining grid cell responses, we may provocatively imagine the following division of labor: in the adult animal, continuous attractor network dynamics are necessary and sufficient for generating spatially periodic grid responses, while multiple temporal oscillators are essential for determining temporal aspects of spiking responses, including phase precession. If a similar composite model were a true reflection of the biology, this would explain the predictive successes of both classes of models. A prediction of this specific composite model, with separable roles for temporal oscillations and recurrent connections, would be that spatially periodic responses should be present even in the absence of temporal oscillations. A second prediction is that phase precession should be evident in the presence of intact feedforward input, bringing external sensory cues that supply the system with information about animal location, even after local recurrent feedback in the entorhinal cortex is disrupted. The ultimate tests of all such ideas and models must come via experimentation. But the work of Remme and colleagues beautifully illustrates how theoretical considerations and numerical computation can unearth basic biophysical constraints, as well as shrink the field of hypotheses into a set that can more manageably be addressed by experiments. Democracy-Independence Trade-Off in Oscillating Dendrites and Its Implications for Grid CellsRemme et al.NeuronMay 13, 2010In BriefDendritic democracy and independence have been characterized for near-instantaneous processing of synaptic inputs. However, a wide class of neuronal computations requires input integration on long timescales. As a paradigmatic example, entorhinal grid fields have been thought to be generated by the democratic summation of independent dendritic oscillations performing direction-selective path integration. We analyzed how multiple dendritic oscillators embedded in the same neuron integrate inputs separately and determine somatic membrane voltage jointly. Full-Text PDF Open Access" @default.
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