Minimal Models of Adapted Neuronal Response to In Vivo–Like ...

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2114 G. La Camera, A. Rauch, H.-R. Lüscher, W. Senn, and S. Fusi f(t) [Hz] 100 50 I [nA] 0 1.8 1.2 0.6 200 300 400 500 time [ms] Figure 6: Time-varying activity of a population of independent, adapting LIF neurons in response to a noisy, broadband stimulus. (Top) prediction of the adapting rate model, equation 3.4 (gray), compared to the simulations of 20,000 neurons (black), as described in section 3. Shown is the activity after a transient of 200 ms. The short horizontal bar indicates a pulselike increase of 1 ms duration in ν 0 , of strength 12ν 0 . The long horizontal bar indicates a 50% steplike increase in ν 0 , during which the inhibitory rate step increases by 0.7ν 0 . (Bottom) Average time course of the stimulus. Neuron parameters as in Figure 1a, apart from τ r = 2 ms. Other parameters (refer to the text): ν 0 = 350 Hz, ν inh = 1.2ν 0 , G ampa,gaba = 200 pA, G nmda = 10 pA, τ ampa,gaba = 5 ms, τ nmda = 100 ms. Bin size of the PSTH: 0.5 ms. Integration time step: 0.01 ms. In Figure 6 we show an example with two fast (τ x = 5 ms) components, one excitatory (AMPA-like), the other inhibitory (GABA A -like), plus a third component mimicking slow (NMDA-like) current (with τ nmda = 100 ms). The latter component is only slowly fluctuating, so that its variance can be neglected (Brunel & Wang, 2001), as in the case of the adaptation current. The output rate was calculated as f (t) = (m ampa + m nmda − m gaba − I ahp , s 2 ampa + s2 gaba ), (3.4) where is equation 2.3 corrected for fast synaptic dynamics (Fourcaud & Brunel, 2002), see section 2.1.5. The excitatory stimulus was of the form ν 0 (1+ (t)), with ν 0 = 350 Hz and (t) a rectified superposition of 10 sinusoidal components with random frequencies ω i /2π, phases φ i , and amplitudes A i drawn from uniform distributions (the latter between 0 and 0.5): [ ] ∑ (t) = A i sin(ω i t + φ i ) . i +
Adapting Rate Models 2115 The maximum value used for ω/2π was 150 Hz. ν I was held constant to 1.2ν 0 . This gives ¯m ampda,nmda = ḡ ampa,nmda ν 0 (1 + (t))τ ampa,nmda , ¯m gaba = 1.2ḡ gaba ν 0 τ gaba , with m x given by equation 3.2, and analogously for s 2 ampa,gaba . The input currents used in the simulations were [I x ] + , where I x evolved according to the OU process, equation 3.1. Each neuron received an independent realization of the stochastic currents. The time-varying population activity was assessed through the peristimulus time histogram (PSTH) with a bin size of 0.5 ms. The model makes a good prediction of the population activity, even during fast transients, as in response to the impulse of 1 ms duration at t = 250 ms and a step increase in ν 0 ,ν inh occurred at t = 400 ms (horizontal bars in Figure 6). The small discrepancies are due to finite size effects (Brunel & Hakim, 1999; Mattia & Del Giudice, 2002), and to the approximation used for the stationary response function. Similar results were obtained with the adapting threshold model (i.e., with I ahp = 0 and equation 3.3 replaced by τ θ ˙θ =−(θ − θ 0 ) + βf ), and for the CB neuron (not shown). It has to be noticed that the condition τ ampa,gaba ≪ τ ∗ , where τ ∗ is the effective time constant of the CB neuron (see equation A.6), is usually more difficult to fulfill, as τ ∗ can reach values as small as a few ms, depending on the input (see, e.g., Destexhe et al., 2001). However, when τ ∗
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