Democracy-Independence Trade-Off in Oscillating Dendrites and Its Implications for Grid Cells

Summary Dendritic 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. We found that the interaction of dendritic oscillations leads to phase locking, which sets an upper limit on the timescale for independent input integration. Factors that increase this timescale also decrease the influence that the dendritic oscillations exert on somatic voltage. In entorhinal stellate cells, interdendritic coupling dominates and causes these cells to act as single oscillators. Our results suggest a fundamental trade-off between local and global processing in dendritic trees integrating ongoing signals.

. Increasing the magnitude of oscillation-generating currents decreases the response to synaptic input (related to Figure 3). Curves show the oscillator phase shift in response to a 1 mV synaptic input for 3 magnitudes of total oscillator current, denoted by scaling factor "I osc ". The magnitude of the oscillation-generating currents can be increased by increasing area a of the oscillating compartment and/or by increasing the conductance densities. Parameter a is included in ε (see Equation S13) and varying the conductance densities scales the PRC amplitude Q (see for example Izhikevich, 2007, chapter 10). The change of the oscillator phase shift Δθ in response to synaptic input is determined by the product of the input, a parameter ε, and the phase response curve, similar to the oscillator response to interdendritic currents (see equation S2), except that ε depends on the synaptic conductance rather than on the intracellular resistance (see equation S13). Parameters of both models were adjusted to generate ongoing oscillations with frequency ~ 8 Hz and a mean membrane potential of ~ −50 mV. Adjusted parameters of the two-variable model were g L = 0.149 mS/cm 2 , g h = 1.35 mS/cm 2 , g NaP = 0.044 mS/cm 2 , E L = −78 mV, and I = 0.9 µA/cm 2 . The full model is described in Fransén et al. (2004), using g L = 0.156 mS/cm 2 , g h(fast) = 1.47 mS/cm 2 , g NaP = 0.087 mS/cm 2 , E L = −78 mV, E Na = 48 mV, and I = 0.9 µA/cm 2 . Figure S3. Optimal dendritic conductances and oscillator size to maximize the phase locking time constant (related to Figure 4 and Supplemental Figure S4). To determine the optimal distance in the stellate cell model between the oscillators and the soma (see Figure 4 and Supplemental Figure S4) we used our analytical framework to compute the phase locking time constant τ lock for a cable with a fixed total length l total of which the oscillators take up a variable fraction α. We consider that the diameter for the oscillating segments and the connecting cable is equal, hence the relative length of one oscillator is α l total /2. Using Equation S14 we computed τ lock as a function of α for a passive cable with standard conductance load ("standard g L ", solid blue line) or high conductance load ("large g L ", dashed blue line), and for those two cables with an additional h-type, restorative conductance ("standard g L + g h ", solid red line, and "large g L + g h ", dashed red line). In order to obtain the maximal τ lock , we also considered a cable with a high conductance load and a hypothetical restorative current that activates very quickly ("large g L + fast restorative current", dashed black line). Note that the active cable current in Figure 3D used the same fast activation. Standard and high conductance load cables used g L = 0.05 mS/cm 2 and g L = 0.3 mS/cm 2 , respectively. The restorative h-type current was based on the I h that was included in the stellate cell simulations shown in Supplemental Figure S4C-E. Active current parameters were computed at V = −50 mV using g h = 1.1 mS/cm 2 and yielded µ = 2.9 with relative density γ R = 1.7 for the cable with standard conductance load, and µ = 0.5 with relative density γ R = 1.1 for the cable with large conductance load. Activation time constant was τ m = 50 msec for the h-type restorative current and τ m = 1 msec for the hypothetical fast restorative current. The total cable length including the two oscillators was l total = 450 µm, diameter d = 0.6 µm, membrane capacitance C m = 1 µF/cm 2 , and intracellular resistivity R i = 200 Ω cm. Oscillator voltage amplitude was = dend 2 V mV and PRC amplitude was Q = 10 ms/mV.
Note that most stellate cell model simulations used α ≈ 75 µm / 225 µm = 0.33 (black arrow). Figure S4. Simulations of stellate cell activity (related to Figure 4). Activity of the stellate cell after simulated explorations failed to produce grid-fields over a wide range of model parameters. Panels A-B-C show gridness mean and standard deviation for 10 explorations of 5 minutes. A: Distance between oscillators and soma was increased from 10 to 200 µm (see Experimental Procedures and Figure 4B) thereby also decreasing the oscillator surface area. Dendrites and soma connecting the oscillators are passive. Parameters are defined by Models 1-7 in Supplemental Table S1. Arrow indicates value used in all other simulations. B: Conductance load on dendrites connecting oscillators was varied by increasing the leak conductance g L from 0.025 to 0.3 mS/cm 2 . Parameters are defined by Models 3 and 8-13 in Supplemental Table S1. Blue and red arrows indicate values used in Figure 4 and panels D-E, respectively. C: A restorative current was added to the soma and dendrites connecting the oscillators. For this we used the same hyperpolarization-activated cation current I h as for the active segments (see Experimental Procedures), varying its density g h from 0 to 1.1 mS/cm 2 . This range corresponds to varying the parameter µ in our mathematical analysis from 0 to 2.9 (see Figure 3D). Parameters are defined by Models 3 and 14-18 in Supplemental Table S1. Blue and red arrows indicate values used in Figure 4 and panels D-E, respectively. D-E: Using parameters that allowed for the most independence between dendritic oscillators, the activity of the stellate cell still failed to produce stable grid-fields and showed strong phase locking throughout the dendritic tree. Our mathematical analysis predicted that adding a large conductance load and restorative currents to the dendritic segments increases τ lock (see Figure 3D and Supplemental Figure S3).
We used the maximal conductance densities shown in panels B and C (red arrows). We also used a distance between the oscillators and soma that was close to optimal for slowing down phase locking (see Supplemental Figure S3 and arrow in panel A). Parameters are defined by Model 19 in Supplemental Table S1. Gridness mean and standard deviation for 10 simulated explorations of 5 minutes was 0.09 ± 0.12. Rate map (D, top panel) and autocorrelation matrix (D, bottom panel) of a single simulation illustrates absence of grid pattern (color coding as in Figure 1 with peak rate 15 Hz). Membrane potentials were strongly correlated between dendrites from different oscillator clusters (E). Note that the large membrane conductances decreased the oscillation period to T = 56 msec.

Dendrites Oscillator
Model  Table S1. Model parameters for simulations of stellate cell activity (related to Figure 4 and Supplemental Figure S4). From left to right, columns give current parameters in soma and dendrites connecting the oscillators ("Dendrites"), oscillator current parameters ("Oscillator"), distance between soma and oscillators ("D osc "), input scaling factor ("K input "), and somatic spiking threshold ("V th "), with conductances in mS/cm 2 , E L and V th in mV, and D osc in µm. Remaining model parameters are described in Experimental Procedures. Simulations in Figure 4 of main text use Model 3 (blue shading). Simulations in Supplemental Figure S4D-E use Model 19 (red shading). Varying parameters of the soma and dendritic segments required adjustments of the oscillator parameters for the model to generate membrane potential oscillations. Note that certain parameter settings require strongly hyperpolarized leak reversal potentials E L to counteract the depolarizing effect of g h and g NaP and maintain the subthreshold oscillations.
Furthermore, the amplitude β of the external input I vel,i (t) was scaled by a factor K input (see Experimental Procedures) to adjust for the model's input resistance. The somatic spiking threshold V th was set such that the model produced mean firing rates of ~ 2 Hz.

Supplemental Experimental Procedures: Derivation of the phase locking time constant
We performed a mathematical analysis to determine how fast phase locking of dendritic oscillators occurs as a function of the oscillator properties and the properties of the membrane segment connecting the oscillators. Consider a system of two identical oscillators with natural frequency f (in Hz) that are coupled via a cable of length l cm, with oscillator i = 1,2 located at x = 0 and x = l, respectively (see Figure 1A). The membrane potential V i (t) (in millivolts) of each dendritic oscillator is described by a sinusoidal function: with dendritic oscillator amplitude dend V (in millivolts), angular frequency ω = 2φ f, phase shift θ i , and resting membrane potential V R . We consider the oscillators are weakly coupled (i.e. the interactions only affect the oscillators' phases). We can then write the changes in the phase shifts of the oscillators as where the positive parameter ε << 1, and Z(t) is the infinitesimal phase response curve (PRC) which is assumed to be identical for both oscillators. It describes the change of the oscillator's phase shift in response to an infinitesimally small and short perturbation at a particular phase (Izhikevich, 2007). Here we consider Z(t) = −Q sin(ωt), where Q is the amplitude of the PRC (in seconds per millivolt). Note that when ω = dend 1 Q V we obtain the PRC of Andronov-Hopf oscillators, the minimal dynamical system to produce the sinusoidal limit cycle oscillations in equation S1. The perturbations p i (t) result from the axial currents that flow between the cable and oscillator i. The passive properties of the cable are determined by a membrane time constant τ (in seconds), a leak reversal potential E L (in millivolts), and a length constant λ (in centimeters), giving the cable an electrotonic length L = l/λ. The cable also expresses a voltagedependent conductance with reversal potential E m . The dynamics of this conductance are determined by a single gating variable m(x,t) with activation function m ∞ (V) and time constant τ m .
The equations governing the membrane potential V(x,t) and the gating variable m(x,t) along the cable are where γ m is the ratio of the maximal conductance of the active current to the passive membrane conductance. In order to determine the perturbations to the oscillators we need to solve equation S3 with the oscillators at the ends of the cable giving the periodically forced end conditions of the cable. For this, we first linearize equation S3 about the membrane voltage V R around which the membrane potential oscillates, leading to the quasi-active approximation for the cable (Sabah and Leibovic, 1969;Koch, 1984). We define U(x,t) as the difference between the oscillating solution and the resting membrane potential V R , i.e. U(x,t) ≡ V(x,t) − V R and we define w(x,t) analogously as w(x,t) ≡ m(x,t) − m ∞ (V R ). The equations describing the quasi-active cable read where Re[z] is the real part of the complex number z and where μτ μ γ ωτ ωτ ωτ We can then show that the perturbation to oscillator i = 1 reads We want to describe the evolution of the phase difference φ(t) = θ 2 (t) − θ 1 (t). For this we first need to determine the phase interaction function H i (φ) that describes the average effect of perturbation p i (t) on the phase of oscillator i over a cycle of period T = 2π/ω. For oscillator i = 1 this interaction function reads φ φ ρ ωφ ξ ν λ and ν is a constant, and where |z| and arg[z] are the absolute value and the angle of the complex number z, respectively. The interaction function H 2 (φ) can be determined similarly. We now have the equation describing the evolution of the phase difference between the two oscillators: The fixed points of this differential equation The parameter ε can be described in terms of cable and oscillator parameters: where R i is the intracellular resistivity (in Ω cm), a is the surface area of one oscillator (in cm 2 ) and C m is the specific membrane capacitance (in µF/cm 2 ). The second equality uses d Rg and τ = C m / g L , where g L is the specific membrane conductance (in S/cm 2 ).
Hence we can write the phase locking time constant as τ τ ρ ξ π λω Considering the voltage at the middle of the cable as the 'somatic' voltage V soma (t) = U(l/2,t), we find the somatic oscillation amplitude for synchronized oscillators:

Supplemental Experimental Procedures: Reduction of the full oscillator model by Fransén et al. (2004) to two variables
We describe how the full model by Fransén et al. (2004) was reduced to the two-variable model given by Equation 7 in Experimental Procedures. The gating of I h was described by a single variable r(V,t) with an activation time constant that was fitted to experimental data from Dickson et al. (2000). The I NaP gating variable n(V,t) was considered to be instantaneous, i.e. n(V,t) = n ∞ (V), since its dynamics are much faster (~ 1 msec) than the oscillation dynamics (see also White et al., 1995;Dickson et al., 2000). Inactivation of I NaP and slow activation of I h were considered constant since their dynamics are much slower than the oscillation dynamics (Magistretti and Alonso, 1999;Dickson et al., 2000). The activation functions n ∞ (V) and r ∞ (V) were identical to those determined experimentally by Magistretti and Alonso (1999) and Dickson et al. (2000). Current densities were constrained by the frequency and amplitude of the resulting subthreshold oscillations recorded at the soma, which needed to be on the order of ~ 8 Hz and 1-2 mV, respectively (Alonso and Llinás, 1989;Alonso and Klink, 1993).

Supplemental Text: Stable grid fields require unrealistic membrane and input parameters to obtain sufficiently slow phase locking time constants
For stellate cells to maintain stable grid fields over many minutes -as observed in in vivo experiments (Hafting et al., 2005) -the phase locking time constant needs to be at least ~ 100 seconds. On the basis of our analytical results we find that this can only be accomplished by decreasing the PRC amplitude more than a factor of 1000, when considering an optimal oscillator size and including active dendritic conductances (see Figure 3 and Supplemental Figure S3). Hence, this requires a 1000-fold increase in the total oscillation-generating current. Using the oscillator model described in Experimental Procedures as a reference, we determined the consequences of such current density increases. The model uses oscillation-generating currents based on experimental data from stellate cells. It has a similar PRC amplitude as the oscillator used in our mathematical analysis (see Supplemental Figure S2) and therefore also requires a 1000-fold increase in the current densities. The total membrane conductance of the oscillator model at the mean oscillation voltage −50 mV is ~ 0.2 mS/cm 2 . Increasing the densities by a factor of 1000 gives very large dendritic conductance densities, e.g. peak g NaP reaching ~ 1 nS/µm 2 , similar to estimates of sodium channel densities in the axon initial segment (Kole et al., 2008). Note, however, that conductances in the axon initial segment are not persistent and so they are mostly inactivated at subthreshold potentials, while the oscillator conductances are persistently activated throughout the oscillation cycle. The increase in membrane conductance of the oscillator model to ~ 200 mS/cm 2 means that the membrane time constant decreases from 5 msec to 5 µsec. This is a 100-times faster than the (to our knowledge) fastest membrane time constants observed in neurons (0.5-1 msec in auditory brainstem neurons, see e.g. Scott et al., 2005). Furthermore, such an increase in current densities also affects the interaction between I h and I NaP , making the dynamics of the oscillator much faster and increasing its frequency substantially. Our simulations showed that a modest 3-fold increase of the oscillator current densities increased the oscillation frequency from 8 Hz (i.e. the oscillation frequency observed in stellate cells) to 18 Hz (Supplemental Figure S4D-E). Another consequence of a very small PRC amplitude is that the synaptic input currents need to be very large to be able to affect the oscillator phase (see Supplemental Figure S1). Making a conservative estimate, the mean total synaptic conductance input at the oscillator can be at most a factor of 100 smaller than the total oscillator conductances to have an effect on the oscillator dynamics. A mean synaptic conductance density of ~ 20 mS/cm 2 requires more than 1000 excitatory synapses per 100 µm 2 , assuming a mean input rate of 10 Hz, synaptic peak conductance of 300 pS, and synaptic time constant of 6 msec (Berretta and Jones, 1996). This is far beyond the observed densities of excitatory synaptic inputs in dendrites of cortical neurons, which are typically in the range of 30-50 synapses per 100 µm 2 (see, for example, Anderson et al., 1994).