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Influence of firing mechanisms on gain modulation

We studied the impact of a dynamical threshold on the f-I curve-the relationship between the input and the firing rate of a neuron-in the presence of background synaptic inputs. First, we found that, while the leaky integrate-and-fire model cannot reproduce the f-I curve of a cortical neuron, the leaky integrate-and-fire model with dynamical threshold can reproduce it very well. Second, we found that the dynamical threshold modulates the onset and the asymptotic behavior of the f-I curve. These results suggest that a cortical neuron has an adaptation mechanism and that the dynamical threshold has some significance for the computational properties of a neuron.


Introduction
Neurons receive thousands of synaptic inputs, which are then transformed into output spike trains. A common assumption in neuroscience is that the firing rate−the average number of spikes per unit time−of a neuron conveys information and is used for computation in the brain. This assumption is supported by a number of experimental studies [1,2,16]. It is therefore very important to study the relationship between the input current and the firing rate (f-I curve) in the presence of background synaptic inputs in order to understand the computation performed by a neuron. Recently, the leaky integrate-and-fire with dynamical threshold (LIFDT) model [5,6,9,11,12,13,15] has been proposed so as to include spike frequency adaptation. Although the LIFDT model is simple, it can reproduce important properties of cortical neurons such as the negative interspike interval correlation [5,15] and the f-I curve of pyramidal neurons driven by a fluctuating current [13]. It is also reported that the spike threshold of a cortical neuron in vivo is not constant but depends on the preceding spike times [10]. Our understanding of the effect of the dynamical threshold on the computational properties of a neuron is lacking. We studied whether the leaky integrate-and-fire (LIF) and LIFDT neuron can reproduce the f-I curve of a cortical neuron [7] and how the dynamical threshold modulates the f-I curve in the presence of background synaptic inputs.

Leaky integrate-and-fire neuron
Here, we briefly introduce the leaky integrate-and-fire (LIF) neuron [12]. The membrane potential V of the neuron obeys the first-order differential equation where C is the membrane capacitance, g L is the leak conductance, V ∞ is the resting potential of the neuron, and I(t) is the input current. When the membrane potential reaches the threshold θ 0 , a spike is generated and we instantaneously reset V (t) to the resetting potential V r . We adopted the parameters of the LIF neuron from Chance et al. [7], where C = 0.

Leaky integrate-and-fire neuron with dynamical threshold
We briefly introduce here the LIFDT neuron [5,6,9,11,12,13,15]. The membrane potential V of the neuron obeys Eq. (1). In the following, we take the parameters C, g L , V ∞ , V r to be the same as in the LIF neuron. When the membrane potential reaches the threshold θ(t), a spike is generated and we instantaneously reset V (t) to the resetting potential V r . The emission of a spike causes the threshold to increase by an amount A θ and then decay to its resting value θ ∞ exponentially: where τ θ is the time constant of the dynamical threshold and t k is the k-th spike time. The sum is taken over all the spikes generated by the neuron up to time t.

Synaptic input
We considered an input current I(t) = m + I s (t), which is the sum of a constant current m and a background synaptic current I s . The synaptic current is described by where g E,I are the excitatory (E) and the inhibitory (I) synaptic conductances, V E,I are their respective reversal potentials, τ E,I are their respective time constants, a E,I are their respective peak conductances, and t E,I k are the k-th spike times of the respective presynaptic neuron. The spike times of the respective presynaptic neuron are generated by an independent homogeneous Poisson process with the same rate γ. The parameters are , and a I = 0.04[µS/cm 2 ], adopted from Chance et al. [7].

Irreproducibility of the f-I curve by the LIF neuron
We compared the f-I curves of a cortical neuron obtained by Chance et al. [7] ( Figure  1A) to those of the LIF neuron ( Figure 1B). We found that the LIF neuron cannot reproduce two main features of the f-I curves of a cortical neuron. The first is the onset of the f-I curve. The onset of the f-I curves of a cortical neuron are linear, whereas those of the LIF neuron are nonlinear. The second irreproducible feature is the asymptotic behavior of the f-I curve. The asymptotic behavior of the f-I curves of a cortical neuron are sublinear, whereas those of the LIF neuron are linear.

Effect of the dynamical threshold on the f-I curve
We investigated the effect of the dynamical threshold on the onset of the f-I curve.
To quantify the nonlinearity of the onset, we fitted the onset of the f-I curve by using the power function where β represents the nonlinearity of the onset, and c 0 , c 1 are the parameters. Figure  2A shows the onset of the f-I curves of the LIFDT neuron with three values of A θ while the synaptic input rate γ is kept fixed. Figure 2B shows the dependence of the onset nonlinearity β on A θ . The onset of the LIF neuron (A θ = 0) is nonlinear−β = 2.0− while the onset of the LIFDT neuron with large A θ is linear-like: β = 1.3. The dynamical threshold linearizes the onset of the f-I curve.
Next, we investigated the effect of the dynamical threshold on the asymptotic behavior of the f-I curve. Here, we derived the asymptotic formulae of the f-I curve. We used the diffusion approximation [13,14], wherein the excitatory and inhibitory synaptic conductances are approximated by the diffusion processes, and then we neglected the time correlation of the synaptic conductances, where ξ E, I (t) are independent Gaussian white-noise processes of zero mean and unit SD. Using Eqs (1), (3), and (4), we obtain where Here, ξ(t) is a Gaussian white-noise process of zero mean and unit SD, and we assume that the synaptic input is balanced, (a E τ E + a I τ I )V ∞ = a E τ E V E + a I τ I V I . We write the firing rate of the LIF neuron (Eq. The firing rate of the LIF neuron can be written as The firing rate of the LIFDT neuron is given by the self-consistent solution [13] of From Eq. (7), we can obtain Thus we can obtain the asymptotic formula of the f-I curve for large m, Figure 3A shows the f-I curve of the LIF neuron, and Figure 3B shows the f-I curves of the LIFDT neuron. In the large m, these f-I curves approach Eqs (6) and (8). The dynamical threshold changes the asymptotic behavior of the f-I curves: the asymptotic behavior of the LIF neuron is linear (f ∼ m), whereas that of the LIFDT neuron is sublinear (f ∼ m 1/2 ).

Reproducibility of the f-I curve by the LIFDT neuron
We compared the f-I curves of a cortical neuron obtained by Chance et al. [7] ( Figure  4A) to those of the LIFDT neuron ( Figure 4B). The LIFDT neuron can reproduce the two main features of the f-I curve of a cortical neuron: the linear onset and the sublinear asymptotic behavior. Our results are summarized in Table 1.