Fig 2 -

(A) When one fixes the input conductance Gin between the S model and a semi-infinite DS model, the magnitude of the input impedance Zin will be higher for the S model for non-zero frequencies. τδ = 10 ms. (B) Increasing τδ of the DS model decreases |Zin| at all non-zero frequencies. Gin = 8 nS. (C) Decreasing the effective dendritic length ℓ while maintaining the same Gin also decreases |Zin| at all non-zero frequencies. τδ = 10 ms.</p

A comparison of two all-to-all networks with excitatory coupling between 5 neurons, one in which all the neurons have SNIC onset and the other in which all the neurons have HOM onset.

Both networks had the same initial phase relations with the final network states shown in (A-B). (A) The SNIC network almost achieves a synchronous network state, though this is weakly stable due to the near-symmetry of the PRC. (B) The HOM network robustly achieves a splay state in which the neurons are phase-locked in which neuron i has a constant phase difference of 1/5 with neuron i + 1. (C) The synchrony measure over time shows that the SNIC network gradually and non-monotonically approaches the synchronous state while the HOM network converges to the splay state far more rapidly. In panels (A-C), time is measured in terms of the number of interspike-intervals (ISIs) of the network spiking rate.</p

The PRCs obtained by simulating detailed multicompartment Purnkinje cell neuron show that the results of the simplified DS morphology are applicable to complicated and realistic dendritic arbours.

Here we increase the input conductance of the multicompartment model by “growing” the dendritic arbour. We can see this switches the PRC shape from SNIC (A) to HOM (B-C) to Hopf (D) as in the DS model earlier. PRCs obtained from the equivalent DS model closely agree with the full morphology at each of the morphological stages.</p

Bifurcation diagram of the Morris-Lecar DS model in terms of (<i>τ</i>_δ, <i>G</i>_in) focussing on dynamical switches.

Here we have taken Iext as the onset current for every value of (τδ, Gin). Points indicate values of (τδ, Gin) we later use for the spike timing response. At the saddle-node loop (SNL) bifurcation, the dynamical spiking type switches from SNIC to HOM, while Hopf onset becomes possible at the BT bifurcation. Increasing τδ above zero increases both and until the BT and SNL bifurcations meet the cusp at the BTC point at . The difference between and decreases as τδ increases, meaning that the range of Gin for which HOM onset exists becomes smaller. For , the SNL bifurcation no longer exists and decreases. Furthermore, the Hopf onset for switches from subcritical to supercritical. Gin for subcritical Hopf onset increases with τδ until , while Gin for supercritical Hopf onset decreases with τδ throughout the whole range.</p

Fig 3 -

(A) Two-parameter local bifurcation diagram in terms of (Iext, Gin) of the Morris-Lecar DS model for various dendritic time constants τδ. At τδ = 0, the DS model is equivalent to an S model with a leak conductance of Gin. Increasing τδ shifts the BT bifurcation and shrinks the Hopf bifurcation curve. (B) shows that initially increasing τδ moves the BT point to higher Gin until it reaches the cusp at the BTC point. Increasing τδ beyond moves the BT point to the lower SN bifurcation curve (dashed) and thus decreases . The Hopf bifurcation emerging from the BT point also switches from subcritical to supercritical when τδ passes the BTC point. (C) shows the Hopf bifurcation emerging more clearly from a BT point when we measure the external input current relative to the higher saddle-node value ISN,high, while (D) shows the Hopf bifurcation when we measure the external input current relative to the lower saddle-node value ISN,low. The saddle-node and cusp bifurcations are depicted in black because they are unaffected by changes to τδ.</p

A comparison of the single-compartment (S) model with a dendrite-and-soma (DS) model with equivalent passive input conductance <i>G</i>_in. In the DS model, the constant input current <i>I</i>_ext is applied to the somatic compartment.

In the S model, Gin is equivalent to the lumped leak conductance GL, while for the DS model, Gin is the sum of the passive somatic (Gσ) and dendritic (Gδ) contributions. When we increase Gin in the DS model, Gσ is kept constant while Gδ is increased.</p

Document containing the equations and parameters of the Morris-Lecar model (Table A); an alternative, relative conductance formulation of the model; active dendrite simulations (Fig A); mathematical derivations of how to find the local bifurcations; procedures for finding the global bifurcations; and the approach and parameters used to calculate the PRCs for reconstructed neuronal morphologies (Table B).

Document containing the equations and parameters of the Morris-Lecar model (Table A); an alternative, relative conductance formulation of the model; active dendrite simulations (Fig A); mathematical derivations of how to find the local bifurcations; procedures for finding the global bifurcations; and the approach and parameters used to calculate the PRCs for reconstructed neuronal morphologies (Table B).</p

Phase-response curves (PRCs) of the Morris-Lecar DS model for various <i>τ</i>_<i>δ</i> and <i>G</i>_in. Values of <i>G</i>_in and <i>τ</i>_<i>δ</i> have been chosen to be around the dynamical switches in the Morris-Lecar DS model.

For example at τδ = 0, nS and for all τδ the cusp bifurcation lies at nS. When ( ms), increasing Gin switches the onset PRC shape first from a symmetric SNIC PRC (A) to an asymmetric HOM PRC (B-C), and later to a Hopf-like PRC (D). Increasing τδ increases the value of Gin at which the SNIC → HOM transition occurs and also decreases the Gin value of the HOM → Hopf transition. For , the PRC transitions straight from SNIC to Hopf-like. DS parameters used ℓ = 5.</p

A comparison of two pairs of neurons with excitatory coupling between, one in which the neurons have SNIC onset and the other in which the neurons have HOM onset.

Both pairs had the same initial phase relation of ψ0 = 0.3 with the final network states shown in (A-B). (A) The SNIC pair almost achieves a synchronous network state, though this is weakly stable due to the near-symmetry of the PRC. (B) The HOM network robustly achieves an anti-phase state in which the neurons are phase-locked to fire half a cycle apart. (C) The PRCs of each neuron in the SNIC and HOM networks. (D) Coupling functions of the SNIC and HOM pairs. Phase-locked states in a pair of neurons exist where the coupling function is zero. In panels (A-B), time is measured in terms of the number of interspike-intervals (ISIs) of the network spiking rate.</p

Compartment-specific inhibition of bAPs and calcium spikes during BAC firing.

<p>A: Effect of inhibition locus on dendritic coincidence signals, when somatic current injection was paired with dendritic excitation (with a delay Δt of 0 ms) to trigger a calcium spike (as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004768#pcbi.1004768.g001" target="_blank">Fig 1D</a>). As in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004768#pcbi.1004768.g003" target="_blank">Fig 3A</a>, the recording site varies along the rows; the site of inhibition varies along the columns. Inhibitory conductance is indicated by the blue traces (inhibition onset was 1.5 ms after stimulation onset.). a: No inhibition. b: Distal inhibition (460 μm, 50 nS) suppressed the calcium spike (I) and thus distal plasticity, but left signaling in the remaining dendritic tree intact (II and III). c: Proximal inhibition (90 μm, 50 nS) affected signaling in the whole apical dendrite by eliminating the bAP (II), and thus the calcium spike (I), but did not affect the bAP in the basal dendrite (III). d: In the presence of a calcium-spike induced somatic burst, one inhibitory pulse was not sufficient to block the propagation of all bAPs into the basal dendrite (III). e: The train of bAPs in the basal dendrite, and thus basal plasticity, was suppressed by four inhibitory conductance changes at a frequency of 75 Hz on the proximal basal dendrite (100 μm, 70 nS) (III), while apical signaling was unchanged (I and II). B: Inhibition of calcium spikes in the distal apical dendrite. Calcium spikes were triggered by coincident bAPs and distal excitation with a temporal separation (Δt) of 0 ms (as in A). Color-coded is the calcium transient in the apical tuft, normalized to its uninhibited value. While the bAP could be modulated by proximal inhibition within a time window of 1 ms (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004768#pcbi.1004768.g003" target="_blank">Fig 3B</a>), calcium spikes were rather insensitive to timing, and were abolished by weak distal inhibition. C: Inhibition of calcium spikes in the distal apical dendrite, when the neuron was driven by excitatory synapses distributed along the apical trunk to represent inputs from oblique dendrites (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004768#sec019" target="_blank">Methods</a>). The EPSPs were paired with distal excitation with a temporal separation (Δt) of 0 ms. As in B, the calcium spike could be modulated, less dependent on timing than the bAP.</p