2.2. Corticothalamic Transfer Functions

The above NFT equations are nonlinear in general. By setting all derivatives in these equations to zero, we find spatially uniform steady states of the system, which are interpreted as characterizing the baseline of normal activity, with firing rates that are in accord with experiment (Robinson et al., 2002, 2004). Linear perturbations from these steady states have been shown to correspond to time dependent brain activity, leading to successful comparisons with numerous experimental phenomena, including evoked responses (Robinson et al., 1997, 2002, 2004, 2005; Rennie et al., 2002; O'Connor and Robinson, 2004; Kerr et al., 2008; van Albada et al., 2010; Roberts and Robinson, 2012; Abeysuriya et al., 2015).

We expand the equations in section 2.1 to first order in perturbations relative to the steady state, denoting steady-state and perturbed quantities by the superscripts 0 and 1, respectively. We then find

To zeroth order, Equations (6)–(8) yield

Equations (10) and (12) can then be used to eliminate the other variables in favor of the Va(0), which yields the nonlinear steady-state equation (Robinson et al., 1998, 2004)

The first order terms in Equations (6)–(8) give

Operation with D_α on both sides of Equation (16), plus use of Equation (14), yields

The gain G_ab(r, t) is the differential response in ϕ_a per unit change in incoming ϕ_b. The net gains of populations of neurons connected serially are denoted by G_abc = G_abG_bc and G_abcd = G_abG_bcG_cd. These gains are parameterized by time, as shown in Equation (20) to represent their dynamics, which is the topic of the next section.

Numerous biophysical processes modulate neuronal coupling strengths, dependent on current or recent activity, including plasticity, long-term potentiation/depression, facilitation, habituation, and sensitization. We employ a general form of modulatory process that can be applied to a broad range of specific mechanisms (Koch, 1999; Rennie et al., 1999; Robinson et al., 2002), which is a form of feedback, whereby presynaptic neuronal activity modulates neuronal gains (postsynaptic involvement is omitted here and postponed to future work), with

where F(t) describes the temporal dynamics of the gain modulation and g_ab is its strength and ⊗ denotes convolution operation. Equation (21) assumes that the perturbations are small enough that a linear equation is a reasonable approximation. Furthermore, the modulation is assumed to be local in space, so the g_ab are constant and the functional form of F(t) does not vary with position or time. For the temporal function of the modulation we use (Robinson et al., 2004)

when t ≥ 0 and zero otherwise to enforce causality. The rate constant η > 0 characterizes the timescale of the feedback process.

The transfer function is the ratio of the output of a system to its input in the linear regime. Either the Laplace or Fourier transform can be used to determine transfer functions; we use the former in time and the latter in space, with the definitions

respectively, where s = −iω = Γ − iΩ is the complex frequency that parameterizes the response e^st.

Replacement of Gab(1) in Equation (18) by Equation (21) yields

Application of (23) and (24) to Equation (26) gives

where L(s) is the reciprocal of the Laplace transform of the operator D_α(t).

Equation (27) expresses first order responses of two types: the first term in the square brackets represents the part of response that would occur without change to the steady-state gains, whereas the second term is the response due to stimulus-induced gain changes of the steady-state activity. In the Laplace domain, the transfer function to excitatory cortical neurons from retina, is (see Babaie-Janvier and Robinson, 2018 for detailed derivation)

where M=Dee(1-GeiLii)-GeeLee, G_ab = G_ab(s) is the Laplace transform of G_ab(t), and

Note that Equation (29) corrects a typographical error in the corresponding equations in Babaie-Janvier and Robinson (2018); however, the paper used the correct equation and their results remain unchanged.

The transfer function fully describes the linear system properties including the linear response to any external signal. Poles of the transfer function yield the characteristic equation of the system, whose roots determine the poles and thus the basic modes into which the system response can be decomposed. Roots of the numerator of the transfer function are the zeros of the system; these frequencies do not pass through the system.

In this study we only explore spatially uniform perturbations (i.e., k = 0) and postpone study of spatial dependences to future work. Such spatially unstructured stimuli have been widely used in visual flicker experiments to probe steady state visual evoked potentials (SSVEPs) (Spekreijse et al., 1973; Herrmann, 2001; Roberts and Robinson, 2012; VanRullen and Macdonald, 2012). It was also shown recently that the spatially uniform contribution dominates the later phases of ERPs (Mukta et al., 2019).