Intrinsic and Synaptic Properties Shaping Diverse Behaviors of Neural Dynamics

Abstract

The majority of neurons in the neuronal system of the brain have a complex morphological structure, which diversifies the dynamics of neurons. In the granular layer of the cerebellum, there exists a unique cell type, the unipolar brush cell (UBC), that serves as an important relay cell for transferring information from outside mossy fibers to downstream granule cells. The distinguishing feature of the UBC is that it has a simple morphology, with only one short dendritic brush connected to its soma. Based on experimental evidence showing that UBCs exhibit a variety of dynamic behaviors, here we develop two simple models, one with a few detailed ion channels for simulation and the other one as a two-variable dynamical system for theoretical analysis, to characterize the intrinsic dynamics of UBCs. The reasonable values of the key channel parameters of the models can be determined by analysis of the stability of the resting membrane potential and the rebound firing properties of UBCs. Considered together with a large variety of synaptic dynamics installed on UBCs, we show that the simple-structured UBCs, as relay cells, can extend the range of dynamics and information from input mossy fibers to granule cells with low-frequency resonance and transfer stereotyped inputs to diverse amplitudes and phases of the output for downstream granule cells. These results suggest that neuronal computation, embedded within intrinsic ion channels and the diverse synaptic properties of single neurons without sophisticated morphology, can shape a large variety of dynamic behaviors to enhance the computational ability of local neuronal circuits.

Keywords: dynamical system; neural dynamic modeling; rebound firing; stability analysis; synaptic dynamics.

Figure 1

UBCs as relay cells between MFs and GCs (A) A typical GC receives four presynaptic inputs from MFs and/or UBCs. (B) Channel properties of mGluR2-dependent rectifier K and T-type Ca currents. (Bi) mGluR2-dependent rectifier K (I_K) steady-state activation curve (top) and its current as a function of membrane potential (bottom). (Bii) T-type Ca steady-state activation and inactivation curves (top) and voltage sensitive time constants τ₁ and τ₂. Note that τ₂ is much larger than τ₁.

Figure 2

Stability analysis of the resting membrane potential of a UBC. (A) Bifurcation diagram based on theoretical analysis of eigenvalue λ at V_rest=-67 mV. The upper part of the bifurcation diagram (red) indicates that the UBC is unstable with positive eigenvalue, and the lower part (blue) is stable with a negative one. (B) Bifurcation diagram simulated by the full model at V_rest=-67 mV. (C) Similar to (B) but for the minimal model.

Figure 3

Stability diagram of a UBC with rebound firing at V_rest = −67 mV. (A) Examples of UBC response in the full model with different combinations of g_T and g_K for unstable state (top), rebound firing (middle), and stable state (bottom) after the offset of a constant inhibitory current. (B) Similar to (A) but with the minimal model. (C) Stability diagram of the UBC in the full model at V_rest =-67 mV showing three regions of unstable state (max_t (V) >-50 mV), rebound firing (max_t (V) ∈ [-60−50] mV), and stable state (max_t (V) <-60 mV) in the parameter space of g_T and g_K. (D) Similar to (C) but with the minimal model.

Figure 4

Subthreshold resonance of a UBC in response to sinusoidal inputs with linearly and non-linearly increasing frequency. (A) Zap current input from 0 to 10 Hz linearly as I_input = I_ampsin(2f(t)t) with f(t) = f_maxt/T at I_amp = 1 pA, f_max = 10 Hz, and T = 10 s. (B) UBC voltage response triggered by the stimulus given in (A) in the full model. (Right) Fourier transform of UBC response showing the resonance of UBC dynamics with a peak frequency around 6 Hz. (C) Similar to (B) but simulated with the minimal model. (D) Zap current input from 0 to 10 Hz non-linearly as $I_{input}=I_{amp}sin(0.35t^3)$ at I_amp = 1 pA. (E,F) Similar to (B,C) showing UBC response and resonance.

Figure 5

UBC response to oscillating current inputs. (A) Response of UBC models to sinusoidal current inputs with 0.5, 2, and 6 Hz frequencies and 10 pA amplitude. (Top) Current inputs of three frequencies. UBC voltage traces obtained from the full (middle) and minimal (bottom) models with fitting curves (green) to compute the phase in degree and amplitude in Hz of UBC responses. The fitting curve was obtained by fitting the instantaneous firing rate of UBC responses. (B) The phase shift of UBC response relative to the input phase in the full (red) and minimal (blue) model as a function of modulation frequency. (C) Similar to (B) but for UBC response amplitude.

Figure 6

UBC response to synaptic inputs with different types of receptors. (A) Receptor dynamics of fast AMPA (AMPAf), slow AMPA (AMPAsl), NMDA, mGluR1 (M1), and mGluR2 (M2) triggered by mossy fiber Poisson spike trains at frequencies of 1, 20, and 50 Hz. (B) Spiking output of UBCs with different combinations of synaptic receptors in response to Poisson stimulation of MF inputs.

Figure 7

UBC response to MF input with a modulated firing rate. (A) (Top) MF spike train input sampled from a modulated sinusoidal firing rate A sin(2πft) + A with 1 Hz frequency and amplitude A = 20 Hz. Vertical ticks indicate spike times. (Bottom) UBC voltage traces simulated with different receptors of AMPAf, AMPAsl, and NMDA, respectively. Fitting curves (green) are sinusoidal functions to compute UBC response modulation in terms of phase shift and amplitude. (B) The phase shift of UBC response relative to MF sinusoidal input with three types of receptors as a function of MF input frequencies. (C) Similar to (B) but for UBC response amplitude.