Fig 1.
A comparison of the single-compartment (S) model with a dendrite-and-soma (DS) model with equivalent passive input conductance Gin. In the DS model, the constant input current Iext 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.
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.
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.
Fig 4.
Bifurcation diagram of the Morris-Lecar DS model in terms of (τδ, Gin) 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.
Fig 5.
Phase-response curves (PRCs) of the Morris-Lecar DS model for various τδ and Gin. Values of Gin and τδ 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.
Fig 6.
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.
Fig 7.
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.
Fig 8.
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.