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

Adapting Rate Models 2117
I ahp =−g N [N] i (see equation 2.1), and √ 2τ ′ is a factor to preserve units (see,
e.g., Rauch et al., 2003). V rest = 0 is the resting potential, C is the capacitance
of the membrane, τ = RC, and R is the membrane resistance. ξ t is a
gaussian process with flat spectrum and unitary variance, 〈ξ t ξ t ′〉=δ(t − t ′ )
(white noise; see, e.g., Tuckwell, 1988, or Gardiner, 1985, for more details).
In a nonadapting neuron, I ahp ≡ 0. In the adapting threshold model (see section
2.2), I ahp = 0 and the threshold θ ≥ θ 0 is a dynamical variable obeying
equation 2.7.
A.2 Quadratic Integrate-and-Fire Neuron. The dimensionless variable
V, to be interpreted as the membrane potential of the white noise–driven
QIF neuron obeys
τdV = (V 2 + µ)dt + σξ t
√
τdt,
(A.2)
where τ is a time constant that mimics the effect of the membrane time
constant, and µ, σ 2 are the average and variance per unit time of the input
current. A spike is said to occur whenever V =+∞, after which V is clamped
to V =−∞for a refractory period τ r . In practice, in the simulations, V is
reset to −50 whenever V =+50. This gives an accurate approximation for
the parameters chosen in Figure 2. On the other hand, in the rate function,
equation 2.4, the actual values used for the integration limits do not matter,
provided they are larger than +10 and smaller than −10 respectively.
A.3 Conductance-Based LIF Neuron. The membrane potential of the
conductance-based LIF neuron obeys
dV =−˜g L (V − V rest )dt + g E (V E − V)dP E + g I (V I − V)dP I ,
where g E,I = τ ḡ E,I /C are dimensionless peak conductances, ˜g L = 1/τ is the
leak conductance in appropriate units (1/ms), V E,I are the excitatory and
inhibitory reversal potentials, and dP E,I = ∑ j δ(t − tE,I
j
)dt are Poisson spike
trains
√
with intensity ν E,I . In the diffusion approximation (dP x → ν x dt +
νx dtξ t ), the equation can be put in a form very similar to equation A.1 (see,
e.g., Hanson & Tuckwell, 1983; Lánský &Lánská, 1987; Burkitt, 2001):
dV =− V τ ∗ dt + µ 0dt + σ 0 (V) √ dtξ t
(A.3)
where
µ 0 = ˜g L V rest + (g E V E ν E + g I V I ν I ) (A.4)
σ0 2 (V) = g2 E (V E − V) 2 ν E + g 2 I (V I − V) 2 ν I
(A.5)
τ ∗ = ( ˜g L + g E ν E + g I ν I ) −1 .
(A.6)