LFP spectral peaks in V1 cortex: network resonance and cortico-cortical feedback

Abstract

This paper is about how cortical recurrent interactions in primary visual cortex (V1) together with feedback from extrastriate cortex can account for spectral peaks in the V1 local field potential (LFP). Recent studies showed that visual stimulation enhances the γ-band (25–90 Hz) of the LFP power spectrum in macaque V1. The height and location of the γ-band peak in the LFP spectrum were correlated with visual stimulus size. Extensive spatial summation, possibly mediated by feedback connections from extrastriate cortex and long-range horizontal connections in V1, must play a crucial role in the size dependence of the LFP. To analyze stimulus-effects on the LFP of V1 cortex, we propose a network model for the visual cortex that includes two populations of V1 neurons, excitatory and inhibitory, and also includes feedback to V1 from extrastriate cortex. The neural network model for V1 was a resonant system. The model’s resonance frequency (ResF) was in the γ-band and varied up or down in frequency depending on cortical feedback. The model’s ResF shifted downward with stimulus size, as in the real cortex, because increased size recruited more activity in extrastriate cortex and V1 thereby causing stronger feedback. The model needed to have strong local recurrent inhibition within V1 to obtain ResFs that agree with cortical data. Network resonance as a consequence of recurrent excitation and inhibition appears to be a likely explanation for γ-band peaks in the LFP power spectrum of the primary visual cortex.

Appendices

Appendix A

1.1 Rate equation and time course of conductance change

Consider the time evolution of conductance for AMPA receptors in a neuron. This can be described by an integral. $m(t) = \int {A_E \left( {t - a} \right)f_E (a)da}$. f _E (t) is $\sum\nolimits_j {\delta \left( {a - t_j^i } \right)}$ where $t_j^i$ is ith spikes from jth presynaptic excitatory neurons and δ(t) is a delta function. A _E (t) is the time course of conductance change of AMPA receptors. The conductance due to GABA_A receptors, n(t) can be described with another function, A _I (t) in the similar way.

When the number of presynaptic neurons is large and the neural network is in an asynchronous state, f _E (t) can be replaced by the mean firing rate of presynaptic neurons, $\overline{f}_E (t)$ and the mean firing rates are determined by the conductances of the presynaptic neurons: $\bar{f}_E (t) = \bar{f}_E \left( {m(t),n(t),I(t)} \right)$ where I(t) is the feed-forward input to the network. Self-consistency requires that m(t) and n(t) should satisfy the following integral equations.

$$ m(t) = \int_{{ - \infty }}^t {A_E \left( {t - a} \right)\bar{f}_E \left( {m(a),n(a),I(a)} \right)da} $$

(A.1)

$$ n(t) = \int_{{ - \infty }}^t {A_I \left( {t - a} \right)\bar{f}_I \left( {m(a),n(a),I(a)} \right)da} $$

(A.2)

Once we have the solution of the above integral equations, the firing rate of neurons is given by the time derivative of m(t) and n(t). But an integral equation is difficult to analyze and we simplify the integral equations with two approximations. First, we transform the integral equations into differential equations assuming that explicit forms of A _E (t) and A _I (t) are given. For example, for an exponential model, $A_E (t) = \exp \left( { - t/\tau_E } \right)/\tau_E$,

$$ \tau_E \frac{{dm(t)}}{{dt}} = - m(t) + \bar{f}_E (t). $$

(A.3)

where τ _E is the decay time constant of the conductance. For AMPA, τ _E is a few milliseconds. while for NMDA, τ _E is 50–100 msec.

For the difference of exponential model (DOE) with time delay, $A_E (t) = \left( {e^{{ - (t - \delta )/\tau_E }} - e^{{ - (t - \delta )/\tau_{{E0}} }} } \right)/(\tau_E - \tau_{{E0}} )$, we find

$$ \tau_{{E0}} \frac{{dm(t)}}{{dt}} = - m(t) + m_0 (t) $$

(A.4)

$$ \tau_E \frac{{dm_0 (t)}}{{dt}} = - m_0 (t) + f_E \left( {t - \delta } \right) $$

(A.5)

where δ is the synaptic delay time, and τ_E0 is the rising time constant of the conductance’s time course. For small τ_E0, Eq. A.4 and A.5 can be merged into one equation.

$$ \tau_E \frac{{dm(t)}}{{dt}} = - m(t) + f_E \left( {t - \delta - \tau_{{E0}} } \right) $$

(A.6)

Another approximation that achieves the same result as Eq. 1.0 is to assume an explicit form of the mean firing rate, $\bar{f}_E (t) = \bar{f}_E \left( {m(t),n(t),I(t)} \right)$. The linear form with rectifying nonlinearity is known to be a good approximation for conductance based model neurons (Shriki et al. 2003). When the firing rate fluctuates with small amplitude, a series expansion of firing rate in terms of synaptic conductance variables gives us a linear form as well (Brunel and Wang 2003). See also Shelley and McLaughlin (2002) for a coarse-grained reduction of an integrate-and-fire, conductance-based neural network model to a rate model.

Appendix B

2.1 Eigenvalue and ResF with feedback from extrastriate cortex

The characteristic equation of the matrix A in Eq. 3.3 is

$$ Det\left( {\lambda I - A} \right) = \lambda^3 - TrA\lambda^2 + R\lambda - DetA = 0 $$

(B.1)

$$ TrA = \tau_E^{{ - 1}} (1 - S_{{EE}} ) + \tau_I^{{ - 1}} (1 + S_{{II}} ) + \tau_{{EE}}^{{ - 1}} $$

(B.2)

$$ R = \tau_E^{{ - 1}} \tau_I^{{ - 1}} \left( {\left( {1 - S_{{EE}} } \right)\left( {1 + S_{{II}} } \right) + S_{{EI}} S_{{IE}} } \right) + \tau_E^{{ - 1}} \tau_{{EE}}^{{ - 1}} \left( {1 - S_{{EE}} - U_{{EF}} U_{{FE}} } \right) + \tau_{{EE}}^{{ - 1}} \tau_I^{{ - 1}} \left( {1 + S_{{II}} } \right) $$

(B.3)

$$ DetA = \tau_E^{{ - 1}} \tau_I^{{ - 1}} \tau_{{EE}}^{{ - 1}} \left( {\left( {1 - S_{{EE}} - U_{{FE}} U_{{EF}} } \right)\left( {1 + S_{{II}} } \right) + S_{{EI}} S_{{IE}} + S_{{EI}} U_{{IF}} U_{{FE}} } \right) $$

(B.4)

The conditions for Eq. B.1 to have eigenvalues with positive real part can be obtained by the Routh-Hurwitz theorem (Weisstein). But here we use a different derivation showing the stability conditions in Eq. 4.2–4 explicitly. The three eigenvalues are either three real numbers or one real number and two complex conjugates. In any case, TrA > 0 if the real parts of all eigenvalues are positive. When one of the real eigenvalues changes in sign from positive to negative, DetA is 0 because DetA is the product all eigenvalues. It means DetA > 0. Finally, when the real parts of two complex eigenvalues change their sign, the three eigenvalues have the form ε ± αi and β where ε,α and β are three real numbers. It is easy to prove the following using the relation between roots and coefficients of the equation.

$$ TrA = 2\varepsilon + \beta, $$

(B.5)

$$ R = \varepsilon^2 + \alpha^2 + 2\varepsilon \beta, $$

(B.6)

$$ DetA = \left( {\varepsilon^2 + \alpha^2 } \right)\beta, $$

(B.7)

$$ TrA * R - DetA = \left( {2a^2 + 2b^2 } \right)\varepsilon + O\left( {\varepsilon^2 } \right) $$

(B.8)

Therefore, the conditions for the real parts of all eigenvalues to be positive are given in the following way.

$$ TrA > 0 = > \tau_E^{{ - 1}} \left( {1 - S_{{EE}} } \right) + \tau_I^{{ - 1}} \left( {1 + S_{{II}} } \right) + \tau_{{EE}}^{{ - 1}} > 0 $$

(B.9)

$$ DetA > 0 = > \left( {1 - S_{{EE}} - U_{{FE}} U_{{EF}} } \right)\left( {1 + S_{{II}} } \right) + S_{{EI}} S_{{IE}} + S_{{EI}} U_{{IF}} U_{{FE}} > 0 $$

(B.10)

$$ TrA*R > DetA = > T\left( {\tau_E^{{ - 1}} \tau_I^{{ - 1}} D + \tau_{{EE}}^{{ - 1}} T + \tau_{{EE}}^{{ - 2}} } \right) - \tau_E^{{ - 1}} \tau_I^{{ - 1}} \tau_{{EE}}^{{ - 1}} S_{{EI}} U_{{IF}} U_{{FE}} - \tau_E^{{ - 1}} \tau_{{EE}}^{{ - 1}} \left( {\tau_{{EE}}^{{ - 1}} + \tau_E^{{ - 1}} \left( {1 - S_{{EE}} } \right)} \right)U_{{EF}} U_{{FE}} > 0. $$

(B.11)

where $T = \tau_E^{{ - 1}} \left( {1 - S_{{EE}} } \right) + \tau_I^{{ - 1}} \left( {1 + S_{{II}} } \right)$ and $D = \left( {1 - S_{{EE}} } \right)\left( {1 + S_{{II}} } \right) + S_{{EI}} S_{{IE}} .$

When TrA ^* R = DetA, the real part of the complex eigenvalues are zero and $R = DetA/TrA = \left( {2\pi \nu } \right)^2 .$

Since is proportional to the inverse of the real part of the eigenvalue, τ _damp  = ∞ on the stability line defined by the Eq. B.11. Equation B.6 shows that $R = DetA/TrA = \left( {2\pi \nu } \right)^2$ in this case. Then the ResF, v obeys

$$ \left( {2\pi \nu } \right)^2 = \left( {T + \tau_{{EE}}^{{ - 1}} } \right)^{{ - 1}} \tau_E^{{ - 1}} \tau_I^{{ - 1}} \tau_{{EE}}^{{ - 1}} \left( {D + S_{{EI}} U_{{IF}} U_{{FE}} - \left( {1 + S_{{II}} } \right)U_{{FE}} U_{{EF}} } \right) $$

(B.12)