Frequency Response to Oscillatory Noisy Inputs

Finally, we compared the coding properties of these multi-compartment neuron models with those of point neuron models. We simulated the multi-compartment models by somatic injection of noisy oscillatory inputs

I(t) =I0+I1cos(2πf t) +σIIsyn(t),

The constant current level I0 was adjusted in simulations to set the mean firing rate ν₀ around 10 Hz. The modulation of the input currents I₁ was taken as 10% of I0. The magnitude of the synaptic noise σ was chosen such that the subthreshold fluctuation of somatic MP had a standard deviation between 3 and 5 mV and the temporal correlation of noisy current has a time constant of 40 ms.

The population response of these model neurons was characterized by the instantaneous firing rate ν(t)averaged over thousands of trials and fitted by the linear response function

ν(t) =ν0+ν1cos(2πf t+φ(f)).

The frequency modulation ν1 of the population firing rate was plotted against the input frequency in Figure 12.10.

1 10 100 1000

0.01 0.1 1

input frequency (Hz) Normalized ν 1

single WB

single WB, gNa x10

multi WB, active dendrites, gNa x10 multi WB, passive dendrites, gNa x10 multi WB, passive dendrites, gNa x20

Figure 12.10: Frequency modulation of Population Firing Rate in multi-compartment models. Asterisk: single-compartment Wang-Buzsaki models with ¯gNa = 300 pS/µm² (black) and ¯gNa = 3000 pS/µm² (magenta). Squares:

multi-compartment Wang-Buzsaki models with active dendrites; g¯Na = 300 pS/µm² at the soma and ¯gNa = 3000 pS/µm² at the AIS. Triangles: multi-compartment Wang-Buzsaki models with large passive dendrites; g¯Na = 300 pS/µm² at the soma, ¯gNa = 3000 pS/µm² (green) and 6000 pS/µm² (red) at the AIS. The temporal correlation of the background synaptic noise τc = 40 ms.

Error bars represent the standard error of the mean (SEM) across independent trials.

From Figure 12.10, the strength of the oscillatory response ν1 in these multi-compartment models showed similar damping at high input frequency as was found in the single-compartment Hodgkin-Huxley-type models. In the presence of synaptic noise with τc = 40 ms, the cutoff frequency was about 50 Hz in both models. For input frequencies f ≥ 200 Hz, the response modulation ν₁ went up significantly when we increased the Na⁺channel density at AIS from 300 pS/µm² to 3000 pS/µm² and 6000 pS/µm² in the passive dendrite models.

On the contrary, a 10-fold increase of Na⁺channel density in single compartment models and active dendrite models has rarely any impact on the modulation amplitude of high frequency response.

12.6 Summary and Discussion

In this chapter, we studied two classes of multi-compartment models of myeli-nated neurons using physiologically constrained parameters. The onset dynamics of somatic APs were analyzed in simulations while systematically varying the length of the AIS and its Na⁺channel density. In models with homogenous chan-nel distribution across soma and dendrites, the somatic AP onset was as smooth as that at the AP initiation sites; In models with large passive dendrites where APs only invaded 5% of the total dendritic membrane, the somatic AP waveforms were strongly shaped by lateral currents in the parameter regime with a distal AIS and a 10-fold or even higher Na⁺channel density at the AIS.

Different from the single compartment neuron models, the input and output sites are spatially separated in real neurons, where soma is considered to be the integration site of synaptic inputs, and AIS is the initiation site of action potentials. How is the neurons’ output function affected by the spatial isolation of soma and AIS in the multi-compartment models?

We have shown above that in most physiological models there are no signifi-cant differences between soma and AIS in AP onset dynamics. Due to the fact that APs are initiated in the AIS, the proximal site of the axon, which is electri-cally tightly coupled to the soma, subthreshold MP fluctuations of these two sites are found to be highly correlated. Our simulations further predict a strong linear correlation of the voltage thresholds at both sites (Figure 12.9). These results demonstrated that it is plausible to predict the spike generation at AIS from the temporal dynamics of the MP fluctuations at soma.

Intriguingly, although dendrites are often treated as the sites of synaptic in-puts, our study suggests that the morphology of the dendritic tree has impact on the somatic AP waveform and the output function of the neuron. The somatic AP onset could reach the observed value of about 20 ms^-1 in models with large passive dendrite if the Na⁺channel density at AIS was as high as 5000 pS/µm². However, so far no direct experimental evidences support the existence of large passive dendritic trees and extremely high Na⁺channel density at AIS. Thus an

overwhelmingly strong lateral current is not very likely to occur in AP initiation dynamics of cortical neurons.

We also computed the response function to noisy input of different frequen-cies in the spatially-extended neuron models. There seemed to be a small but significant improvement of the high frequency response in these models with pas-sive dendrites compared to the point neuron model and the models with active dendrites. Nevertheless, all the multi-compartment models we have investigated exhibited a power-law-like decay of the modulation amplitude for input frequency f >50Hz, contradicting to the experimental observation of undamped responses up to 200 Hz in cortical neurons (Köndgen et al., 2008).

Discussion and Conclusion of