Dendritic signals command firing dynamics in a mathematical model of cerebellar Purkinje cells

Abstract

Dendrites of cerebellar Purkinje cells (PCs) respond to brief excitations from parallel fibers with lasting plateau depolarizations. It is unknown whether these plateaus are local events that boost the synaptic signals or they propagate to the soma and directly take part in setting the cell firing dynamics. To address this issue, we analyzed a likely mechanism underlying plateaus in three representations of a reconstructed PC with increasing complexity. Analysis in an infinite cable suggests that Ca plateaus triggered by direct excitatory inputs from parallel fibers and their mirror signals, valleys (putatively triggered by the local feed forward inhibitory network), cannot propagate. However, simulations of the model in electrotonic equivalent cables prove that Ca plateaus (resp. valleys) are conducted over the entire cell with velocities typical of passive events once they are triggered by threshold synaptic inputs that turn the membrane current inward (resp. outward) over the whole cell surface. Bifurcation analysis of the model in equivalent cables, and simulations in a fully reconstructed PC both indicate that dendritic Ca plateaus and valleys, respectively, command epochs of firing and silencing of PCs.

Figure 1

Local models of dendritic and somatic dual electroresponsiveness (see text). (A) Plateaus triggered by a depolarizing current pulse (100 ms, 130 nA/cm², blue traces) with different I_dc values (nA/cm²). (B) Ca spiking (I_ϕ = 10³ nA/cm². (C) Bifurcation diagram of the dendritic model. R: resting state; M: middle branch; P: plateau; Ω: bistability zone; H: Hopf bifurcation; (u)sLC: (un)stable limit cycles; sn: stable node; s: saddle. (D) Valleys triggered by a hyperpolarizing current pulse (100 ms, −130 nA/cm²) with different I_dc values. (E) Bifurcation analysis of the isolated soma model. (E₁) Soma bistability (I_ϕ = −150 nA/cm² and 150 nA/cm²). (E₂) Bifurcation diagram. F, Fl, Hm: fold, flip and homoclinic bifurcations. Inset: Spontaneously resetting Na plateau with I_dc lying between Hm and Fl; scale bars: 20 mV/ 200 ms. (E₃) Magnification of the rectangle shown in E₂. (E₄) f-I relation.

Figure 2

Infinitely long dendrite. (A) Ca spikes propagate. (A₁) Bifurcation diagram of traveling Ca-spike solutions. Propagating fronts between R and P subthreshold to Ca spikes are found in Ω (♦); (•), emergence of traveling Ca spikes trains. (A₂) (Left) Enlarged view of the “▪” box. (Right) Samples of unitary traveling Ca spikes from branches S_1-2 and S_5-6 (branches S_3-4 are unstable). (A₃) Plateau and valley duration (d) in S₁ and S₆ versus I_dc. (B) R ↔ P fronts propagate. (B₁) Enlarged view of the “♦” box; uf: unstable fronts, sf: stable fronts. (B₂) Examples of RP₁ and PR₁ fronts. Dashed lines indicate the M state voltage. (C) Plateaus and valleys cannot propagate. V (green) and V_ξ (red) nullclines and stable and unstable manifolds (blue) of steady states of the 2D simplified model rewritten in the ξ reference frame, for different I_dc values (nA/cm²): 15 (C₁) and 33 (C₂) inside Ω, and 0 (C₃) and 45 (C₄) outside Ω. In C₁, orange and yellow unstable manifolds are for θ = 0.45 and 0.55 cm/s, and the blue one is for 0.497 cm/s. θ = 0.23 in C₂ and 1.00 in C_3-4.

Figure 3

Electroctonic equivalent cables. (A) Responses of the somatofugal cable to CF inputs. Starting from either R or P, the CF (g_CF = 1.2 μS uniformly distributed over smooth dendrites) triggers a burst of Ca spikes (s), followed by a plateau (p) or a valley (v) and convergence to R or P. Samples of RsR, RspR, RsP (A₁), PspR and PsP (A₂), and PsvP and PsR (A₃) responses are shown with the corresponding I_dc value (nA/cm²). (A₄) Duration (d) of plateaus and valleys versus I_dc. (B) Responses of somatopetal reduced cables to distal inputs. Membrane potential of the farthest dendritic point from the soma (black) stimulated by EPSC/IPSC trains, and the soma (red). Green: Membrane current at the stimulation site. Blue: Synaptic currents. Five (B₁), eight (B₂), and 10 (B₃) EPSCs with g_PF = 0.1 μS/cm²; I_dc = −25 nA/cm². Six IPSCs (B₄) with g_SC = 0.5 μS/cm² and I_dc = 45 nA/cm². (C) Duration of plateaus (C₁: six EPSCs; C₃: g_PF = 0.5 μS/cm²; I_dc = −25 nA/cm²) and valleys (C₂: 6 IPSC; I_dc = 45 nA/cm²) as a function of synaptic parameters. S (absicca): Stimulated membrane surface.

Figure 4

Dendritic control of SS firing. (A₁) Bifurcation analysis of the somatofugal equivalent cable with I_dc (injected into the active soma). Thick line: Lower bound of the somatic limit cycle corresponding to SS firing (orange: stable, green: unstable). Thin lines: Dendritic voltage at electrotonic distance L = 0.15 from the soma (the spike amplitude is ∼0 at this distance). SS firing arises, as in the isolated soma, from a fold (F) bifurcation and is substituted for by Na-Ca bursting at a torus (T) bifurcation. (A₂) F-I_dc curves of SS and Na-Ca bursts (see text). (A₃) Na-Ca bursting (I_dc= 6×10³ nAcm⁻²; black: soma voltage; red: dendritic voltage). (A₄) The same as A₃, with g_c raised to 15 μS. (B) Dendritic control of SS firing by CF (see text). (B₁) I_dc = −25 nA/cm². (B₂) I_dc = 45 nA/cm². (B₃) I_dc = 25 nA/cm². (C) Dendritic control of SS firing by PF EPSCs and SC IPSCs (see text). (C₁) Volley of six EPSCs (50 Hz), g_PF = 0.5 μS/cm², I_dc = −25 nA/cm². (C₂) Volley of six IPSCs (50 Hz), g_SC = 0.5 μS/cm², I_dc = 45 nA/cm². (C₃) Same PF volley as in C₁ followed by the same SC volley as in C₂ with I_dc = 25 nA/cm².