In this section, we summarize the key findings presented in Cooray et al. (2024). The cortical surface is modelled as a thin sheet, just a few millimetres thick, covering an area of 1-2 \(m^2\) in the human brain. Within this sheet, neural units span cortical layers, with connections either orthogonal to the surface (intrinsic connections) or running along the sheet (extrinsic connections), as depicted in Fig. 1. The cortical field dynamics emerge from multiple bi-layers of excitatory and inhibitory neural units, where balanced activation results in oscillatory activity. This balance allows the field to be expressed as a complex vector, with one complex component per bilayer. The dynamics are governed by Eq. (1). The first term on the right-hand side arises from the excitation-inhibition balance, leading to oscillations. The second term introduces perturbations to these oscillations when the interaction is weak. The matrix \(\textbf\) represents the connectivity gain between different layers and between points \(\textbf\) and \(\textbf_0\) on the cortical surface. The vector field \(\textbf\), in conjunction with \(\textbf\), determines how field activity at point \(\textbf\) influences the field activity at point \(\textbf_0\). For a neural field, \(\phi \), we have:

$$\begin \partial _ \varvec = -i\varvec + \iint _ \textbf(\textbf_0,\textbf) \, \textbf(\varvec (\textbf_0),\varvec(\textbf)) \, \text A \end$$

(1)

The integral covers a disc-shaped region around \(\textbf_0\) on the cortical surface, where local connections to \(\textbf_0\) are assumed to be relevant. The literature suggests that this region corresponds to the size of a few cortical columns, with a diameter ranging from 0.1 to 1 mm (Katzner et al., 2009). The neural field descriptions derived from the dynamical system in Eq. (1) have been extensively discussed in the literature (Cook et al., 2022; Jirsa, 2009; Coombes & Owen, 2005). We will assume that our system dynamics remain close to an equilibrium state and focus on perturbations around this state, implying that variations in neural field strength are small. Additionally, we will assume a long-wavelength approximation, excluding the high-frequency aspect of the spectral activity. Spectral analysis has been extensively studied in the literature, particularly in relation to the relationship between frequency and power in LFP, intracranial EEG, and scalp EEG. This relationship is well-modeled by an inverse algebraic function, \(\frac\) , where the beta exponent has been reported to vary between 0.5 and 4 across different frequency ranges (Bedard et al., 2006; He, 2014; Logothetis et al., 2007). The underlying cause of this relationship remains a topic of debate, with some suggesting that the exponent differs for low- and high-frequency activity due to factors ranging from transmitter-receptor dynamics to signal transmission through neural tissue. In Cooray et al. (2024), a dynamic explanation for the inverse power law was proposed, attributing it to the equipartition of energy. In human brain recordings, frequencies below 100 Hz are usually considered to satisfy this approximation (Miller et al., 2009). In Sections 2.2–5, we will explore the invariant structures of the neural field and their implications for dynamics. We will further generalize Eq. (1) to investigate connectivity between neural units and the related invariant structures, enabling us to propose an interaction between cortical connections and neural field activity.

2.1 Dynamics for a single cortical layer

The dynamics of a single bi-layer are defined in Eq. (1), and under a long wavelength approximation, these dynamics can be expressed using a partial differential equation; see Cooray et al. (2024) for further details.

2.1.1 Dynamics of weak connectivity

It is important to note that we are perturbing around a zero neural field state with weak connection strength, W.

$$\begin \begin |\phi |&\ll 1 \\&\\ |\textbf|&\ll 1 \\ \end \end$$

(2)

The S-field is approximated by the first-order expansion of a sigmoid function, specifically \(\tanh \), see Eq. (1). To ensure that the connection parameter remains real, we define the connection as \(-2iW = U\). While imaginary connection gains might seem unusual, they result from using a complex-valued dynamical equation and represent cross-connections between excitatory and inhibitory cells. This network configuration is responsible for generating oscillatory activity. By including only linear terms of U and expanding the integral using a Taylor series for the integrand, we obtain a wave equation.

$$\begin \begin \partial _^ \phi&= -i\partial _\phi + \partial _\iint _A W\phi \ \, \textrmA\\ \partial ^_ \phi&= -\phi + \iint _A U \phi \, \textrmA \\ \partial ^_ \phi&\!=\! -\phi \!+\! \iint _A \left( U \phi \right) |_ \textrmA \!+\! \iint _A \frac\partial ^_\left( U\phi \right) |_ (r\!-\!r_0)^ \, \textrmA\\&= -\phi + Ua\phi +\partial _^\left( U\phi \right) b\\&= - \left( 1-Ua-b\partial _^U\right) \phi +Ub\partial _^\phi \\ \end \end$$

(3)

a and b are defined using integrals over the disc, A.

$$\begin \begin a&= \iint _A \, \textrmA\\ b&= \frac\iint _A r^ \, \textrmA\\ \end \end$$

(4)

The dynamical equation governing the system is given by Eq. (5). Note that Eq. (3) does not include first-order derivatives of U, reflecting the assumption that there is no intrinsic directionality of the connections on the cortical surface. This assumption can be relaxed, and there is some empirical evidence to do so; however, we will not include an analysis of such a structure as the derivations would be too cumbersome to follow.

$$\begin \begin \partial ^_ \phi - Ub\partial _^\phi&= - \left( 1-Ua-b\partial _^U\right) \phi \\ \end \end$$

(5)

Equation (5) is the Klein Gordon field (wave equation with mass), where the mass, m, and speed, c, terms are defined as follows which has been studied in Cooray et al. (2024).

$$\begin \begin m^c^&= \left( 1-Ua-b\partial _^U\right) \\ c^&= Ub\\ \end \end$$

(6)

For the moment, we will assume that U(r, r) is constant across the surface while approximating the system dynamics using U and its higher-order derivatives (which we assume are even-dimensional). Additionally, we can derive the time derivatives of the dynamical system by integrating over the retarded time rather than the evaluation time simplifying the derivation as shown in Eq. (7).

$$\begin \begin&\iint _ U(t_,r) \phi (t_,r) \, \textrmA = Ua\phi + \iint _A \frac\\ &\left[ -\frac\partial ^_\left( U\phi \right) |_+\partial ^_\left( U\phi \right) |_\right] (r-r_0)^\textrmA \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad = \left( Ua+b\partial _^U\right) \phi + \iint _A \frac\\ &\left[ -\frac\partial ^_\left( \phi \right) |_+U\partial ^_\left( \phi \right) |_\right] (r-r_0)^\textrmA \\ \end \end$$

(7)

We can combine Eqs. (3) and (7) to give an integral equation which will be equivalent to Eq. (3) where, \(A'\) indicates that the integral is done over the retarded time. Equation (8) is a concise description of the neural field dynamics.

$$\begin \begin \iint _ U \phi \, \textrmA'&= \phi \\ \end \end$$

(8)

We now take a short detour–describing the Lagrangian formalism–as this will play an important part later in Section 2. The neural field dynamics (Eqs. (3) and (6)) has an associated Lagrangian density which will be a mixture of quadratic functions of the neural fields and its derivatives, Eq. (9).

$$\begin \mathcal _0(\phi ,\phi ^,\partial _ \phi ,\partial _ \phi ^) = \partial _ \phi ^ \partial _ \phi - c^ \nabla \phi ^ \nabla \phi + m^c^ |\phi |^ \end$$

(9)

The notation can be simplified further–using standard terminology from classical field theory–by introducing a metric, \(\textbf\). At present we are assuming the Minkowski metric which will not add any new information to the dynamical equation we have derived. We further use shorthand notation from physics, were repeated up and down indices (co-variant and contra-variant indices) are summed over; roman indices run from 1 to 2 and Greek indices from 0 to 2. The Lagrangian density is then given by Eq. (10).

$$\begin \mathcal _(\phi ,\phi ^,\partial _ \phi ,\partial _ \phi ^) = g^\partial _ \phi ^ \partial _ \phi + m^c^ |\phi |^ \end$$

(10)

The metric defined by the connection U is actually a pseudo-metric as it contains both positive and negative terms along the diagonal, Eq. (11).

$$\begin \begin }^ = \begin1 & 0 & 0\\ 0& -Ub& 0\\ 0& 0& -Ub\end \end \end$$

(11)

The dynamical equation of the neural fields (Eq. (6)) is given by the variation of the neural field in the Lagrangian density. Varying the complex conjugate of the neural field, \(\phi ^*\), will give Eq. (6) and varying the neural field will give the complex conjugate of Eq. (6). This variation is given by the Euler-Lagrange equations.

Fig. 2

Schematic figure of the bi-layer cortex with excitatory and inhibitory neural units. A. The bilayer has constant connections between the two layers, tuned to create oscillations, The connections are drawn as black circles between the layers. B. The bi-layer requires a compensatory variable connectivity to maintain oscillations. These variable connections are drawn as red arrows between the layers

2.1.2 Dynamics of non-weak connectivity

The analysis in Section 2.1.1 assumed that the interactions are weak; however, it is possible to drop this assumption. A system with strong intrinsic connections can be shown to have a non-zero stable state in contrast to the dynamics analysed in Section 2.1.1 but the overall field equations can be shown to be similar. To formalise the above, we will start by defining our model using Eq. (12). The main difference between Eqs. (2) and (12) is the strong intrinsic connection we have included in the form of a Dirac function \(\delta \), i.e. the intrinsic connectivity is of much greater strength than the extrinsic and is modelled using a point like function.

$$\begin \begin \partial _\phi&= -i\phi + \int T(\phi )W \\ T(\phi )&= i\left( \alpha \phi +\beta \phi |\phi |^2\right) \delta (\textbf) +i \gamma \phi \\ W&= |W| \end \end$$

(12)

The dynamical equation will be given by Eq. (13) and can be shown to have a set of non-zero stable states.

$$\begin \begin \partial _\phi&= -iW\left( 1-\alpha + a\gamma + \beta |\phi |^2\right) \phi + iW\gamma b \nabla ^2\phi \\ \end \end$$

(13)

Perturbing Eq. (13) at any of these stable states will give (in a long wavelength approximation) a dynamical equation for a complex field (Eq. (14)).

$$\begin \begin \partial _^2\left( \psi \right)&= 2W\beta \gamma b |\phi _0|^2\nabla ^2 \left( \psi \right) \\ \partial ^2\psi&= 0\\ c^2&= 2W\beta \gamma b |\phi _0|^2\\ \end \end$$

(14)

This state will have a different metric to what was defined in Section 2.1.1 (as the speed of propagation of the neural fields is different) and moreover, there is an absence of a mass term.

$$\begin \begin g^\partial _ \partial _\psi&= 0\\ \end \end$$

(15)

The metric, \(\textbf\), is given in Eq. (16).

$$\begin \begin }^ = \begin1 & 0 & 0\\ 0& -2W\beta \gamma b |\phi _0|^2& 0\\ 0& 0& -2W\beta \gamma b |\phi _0|^2\end \end \end$$

(16)

The neural-connectivity field coupling as discussed in Sections 2.2–4 will be different for the neural field derived in 2.1.1 and that in Section 2.1.2. Equation (5) is Lorentz invariant while Eq. (15) will be conformally invariant (which includes Lorentz invariance). The neural field described in Section 2.1.1 will model small oscillation activity while that of 2.1.2 can be used to model activity showing high amplitude oscillations, e.g. alpha oscillations in human EEG (Nunez, 1974).

2.2 Single layer cortical dynamics with variable intrinsic connection

In deriving the above equations we have assumed that the connection U(r, r) in the cortical bi-layer is isotropic and does not change with time. However, there is compelling empirical evidence that cortical connections change along the surface and that they are time dependent, see Section 3.

In this section we will analyse the effect of spatial and temporal variation of the intrinsic connection. See Fig. 2 for a schematic how variations in the intrinsic connection strength can be compensated by a variable connection strength to retain the oscillation as will be analysed in this section. In brief, the connectivity U above will be modulated using a space and time dependent phase, \(\theta \).

$$\begin \begin U&\rightarrow e^)}U\\ \end \end$$

(17)

The modulation of U given by \(\theta \) is a function on the cortical surface that varies with time and space. Including the modulated connection term into our dynamical system (Eq. (8)) gives us Eq. (18).

$$\begin \begin \phi&= \iint _ e^-\theta )} U \phi ^ \, \textrmA'\\ \end \end$$

(18)

Expanding the integral and taking into account the retarded time gives the following equation.

$$\begin \begin \phi&= \iint _ Ue^-\theta )}\phi ^\, \textrmA'\\&= aU\phi +b\partial _^2U\phi -e^g^\partial _\partial _ \left( e^\phi \right) \\ \end \end$$

(19)

Equation (19) is a constraint on the time-space dependent connectivity field, \(e^\), and the neural field, \(\phi \). We will define a new variable, \(\textbf\), and a perturbation constant, \(\epsilon \), (to use standard nomenclature from classical field theory).

$$\begin -\epsilon A_=\partial _\theta \end$$

(20)

Combining Eqs. (19)-(20)–and after some manipulation–we get an expression for the interaction between the cortical and connectivity field in the format of a classical gauge field. We will define the connectivity field to be that of, \(\textbf\), and the neural field to be equal to \(\phi \).

$$\begin \begin -(1- aU\phi -b\partial _^2U)\phi&= g^\left( \partial _\partial _\phi +\left( -i\epsilon \partial _ A_\right. \right. \\ &\left. \left. -\epsilon ^2 A_ A_\right) \phi -2i\epsilon A_\partial _\phi \right) \\ -m^c^\phi&= g^\left( \partial _-i\epsilon A_\right) \left( \partial _-i\epsilon A_\right) \phi \\ \end \end$$

(21)

The terms in the parenthesis can be seen as derivative terms and are defined in classical field theory to be the covariant derivative.

$$\begin \begin D_&= \partial _-i\epsilon A_\\ \end \end$$

(22)

Using the terminology defined we can succinctly rewrite the dynamics as shown in Eq. (23).

$$\begin \begin g^D_D_\phi + m^c^ \phi&= 0\\ \end \end$$

(23)

Note that the modulation of the connection, \(\theta \), does not give rise to solenoidal vector fields when the derivative is taken (as it is taken to be a real scalar). However, we can allow for more variability in the connection giving us \(A_\)-terms that are mixtures of gradient and solenoidal fields. Returning to Eq. (18) we define the modulation of the connection U as follows, where \(\textbf\) is at the centre of the integrating domain.

$$\begin \begin \iint _A e^)}U\phi \ \, \textrmA&\rightarrow i\iint _A e^, z,z^*)-f(\textbf, 0,0))}U\phi \ \, \textrmA\\ \end \end$$

(24)

When we derive the dynamic equations we get a similar expression but with the connectivity field defined using Eq. (25).

$$\begin \begin -\epsilon A_&=\partial _f(\textbf_0, z,z^*)\\ \end \end$$

(25)

Note that this is not a gradient field as f can be seen as a two dimensional vector. Equation (23) defines the interaction between the neural field and itself and the interaction between the neural and connectivity field; however, the interaction between the connectivity field and itself is not defined. The dynamics can be modified to include an interaction term for the connectivity field by including a kinetic interaction. This is done without too much complication by modifying the Lagrangian of the dynamical equation. The Lagrangian density for the unmodified system is given in Eq. (26) corresponding to the dynamical equation (Eq. (23)).

$$\begin \mathcal _(\phi ,\phi ^,\textbf \phi ,\textbf \phi ^) = g^D_ \phi ^ D_ \phi + m^|\phi |^ \end$$

(26)

To this Lagrangian density we will add a kinetic term for the connectivity field which will give us squared differential terms in the dynamical equations of motion. The Lagrangian density for the connectivity self-interaction is given in Eq. (27).

$$\begin \mathcal _(A, \partial A) = \frac F_F^ \end$$

(27)

Where \(F_\) is given in expression Eq. (28).

$$\begin \begin F_&= \partial _A_-\partial _A_\\ \end \end$$

(28)

The generalised Lagrangian density with interaction terms and self-interaction terms between neural fields and connectivity fields are given (Eq. (29)) as the sum of the Lagrangian densities (Eqs. (26) and (27)).

$$\begin \mathcal (\phi ,\phi ^,\textbf \phi ,\textbf \phi ^, A, \partial A) = g^D_ \phi ^ D_ \phi + m^|\phi |^ + \frac F_F^ \end$$

(29)

Fig. 3

Schematic figure of a lattice where the connection strength is reflected by the length of the connections. This lattice can be projected onto a surface such that each connection becomes of the same length as shown on the curved hyperbolic surface. This surface can be estimated from the connection strengths as discussed Section 2.3

Before we develop the theory any further, we will pause to discuss some important aspects of the above dynamical equation.

1)

The dynamics of the connectivity field will persist even when the neural field is 0. This partly surprising aspect of the theory finds support in experimental evidence where self-interacting connectivity fields have been described (Minerbi et al., 2009; Shimizu et al., 2021).

2)

The Lagrangian density that was derived (Eq. (29)) is the simplest one that allows for the dynamical equation to be invariant to phase transformations (as evidenced in data from cortical tissue) and with a dynamical self-interaction term for the connectivity field.

3)

The mode of interaction is causal on the speed of propagation of the neural and connectivity field between different events. The first term on the right-hand side of Eq. (29) mediates the causal dependency of the neural fields and the third term that of the connectivity field. If these fields were not included (e.g. if the Lagrangian density of Eq. (26) was used) we would have a situation where neural fields would create a disturbance in the connectivity field, which then would transmit infinitely quickly causing secondary changes in the neural field. The result would be neural fields propagating (indirectly) infinitely quickly, this again contrary to experimental findings.

4)

The equations are completely analogous to that of electromagnetism. Notice that there is no clear physical relation between classical electrodynamics and neural-connectivity field theory except the co-existence of phase invariance of oscillatory activity in both types of fields. However, in contrast to electromagnetism where the gauge field (the equivalent of the connectivity field) does not have an empirical counterpart (at least for classical fields), it does so for cortical dynamics. The empirical counterpart is given by the actual connections between the neural units via axons, dendrites, synapses and possibly through the astrocytic scaffolding in which the neural units are embedded.

The main characteristics of cortical activity that allowed us to derive Eq. (29) rest on the following three criteria (i) cortical activity seems to fine tune the excitatory and inhibitory dynamics allowing for phase invariance (cf., excitation-inhibition balance). (ii) The existence of a neural field interacting with a connectivity field and (iii) a minimally complicated dynamical equation supporting point 1 and 2. Point (i) is supported by the oscillatory dynamics seen in data and also finds support in the idea of SOC, where the dynamics tend to a state of invariance, in the current setup a phase-invariance (SU(1)-invariance). Point (ii) does again have some empirical evidence. Point (iii) is mainly for computational reasons but is important, especially in situations where model parameters might be inferred from data.

2.3 Cortical surface geometry and the neural-connectivity field

The geometry of the cortical surface–of relevance to the Neural-Connectivity field–is not determined by the physical geometry of the cortical tissue, but by the interaction of the fields on the surface on which they are defined. The geometry will be defined by the extrinsic connectivity. See Fig. 3 for a schematic depiction of this.

In the following, we consider the mathematical constraints that fulfill a set of plausible geometrical assumptions about the neural connectivity model. 1) The cortical sheet is locally a smooth two-dimensional surface, i.e., a 2 dimensional differential manifold. 2) The concept of distance and angles exist which would require a Riemannian (or pseudo-Riemannian) manifold. 3) Movement of the neural fields on the surface is constrained by the intrinsic curvature of the surface without any external forces or fields required.

To model the interaction between the extrinsic connectivity and the geometry we will modulate the afferent connection (U) using the expansion in Eq. (30).

$$\begin \begin \iint _ U \phi \, \textrmA&\rightarrow f( U,\textbf,\textbf ,\phi ) \\ \end \end$$

(30)

The function \(f( U,\textbf,\textbf,\phi )\) is defined in Eq. (31) where the connectivity consists of a scalar, a vector and a matrix component. These act on the neural field, the derivative and a second derivative of the neural field.

$$\begin \begin f(U,\phi )&= aU\phi - V^\partial _\phi - W^\partial _\partial _\phi \\ \end \end$$

(31)

It can be shown that the geometry or equivalently the metric induced by the above second order expansion of the neural field and its derivatives is, in the simplest case, given by the following relations for the coefficients \(\textbf\) and \(\textbf\).

$$\begin \begin W^&= g^\\ V^&= \frac}\partial _\left( \sqrtg^\right) \\ \end \end$$

(32)

The function f is given in Eq. (33) for the simplest geometrical setting.

$$\begin \begin f(g^,\phi )&= aU\phi - \frac}\partial _\left( \sqrtg^\partial _\phi \right) \\ \end \end$$

(33)

The dynamical equation is given by the following expression which simplifies as follows.

$$\begin \begin \phi = \iint _ U \phi \, \textrmA&= aU\phi - \frac}\partial _\left( \sqrtg^\partial _\phi \right) \\ 0&= (1-aU)\phi + \frac}\partial _\left( \sqrtg^\partial _\phi \right) \\ \end \end$$

(34)

The last equation is the Klein Gordon field on a geometry with metric defined by \(g^\) and the Levi-Civita connection. The expansion for the connection function is constrained by the two dependent terms V and W. This is necessary to define both the metric and the unique torsion free connection (the Levi Civita connection). With this we have fulfilled constraints 2 and 3 above. With this geometry the movement of waves can be defined by the metric itself and its derivatives (as can be seen in Eq. (34)). The curvature of the surface will bend the waves according to Eq. (34). Without the constraint between V and W we would have defined both a metric and connection which would in general also create a torsion “force” on the surface. Constraint 3-i.e. the absence of external forces bending the fields as they propagate over the surface-can be relaxed but the dynamical theory will then need further elaboration. This is something we will not further analyse in this paper.

Equation (34) gives the interaction of the neural field with itself and between the connectivity field (i.e. the metric) and the neural field. Similar to the analysis in Section 2.2. we can generalise the equation of motion to include the interaction of the connectivity field (the metric) with itself. This is easiest done by analysing the Lagrangian density of the dynamics. As it stands, the density is given in Eq. (35).

$$\begin \mathcal (\phi ,\phi ^,D_ \phi ,D_\phi ^) = \sqrt\left( g^D_ \phi ^ D_ \phi + m^|\phi |^\right) \end$$

(35)

The simplest kinetic term that could be added to the Lagrangian is the Ricci scalar R (Sotiriou & Faraoni, 2010). The value of the Ricci scalar at a point on the surface will measure the dispersion of straight lines originating from that point, or equivalently the curvature of the surface. The Ricci scalar is defined using the metric, \(\textbf\), as shown in Eq. (36).

$$\begin \begin \Gamma ^_&= \fracg^\left( g_+g_-g_\right) \\ R^\rho _&= \partial _\Gamma ^_ -\partial _\Gamma ^_+\Gamma ^_\Gamma ^_-\Gamma ^_\Gamma ^_\\ R&= g^R_ = R^\lambda _ \\ \end \end$$

(36)

The modified Lagrangian with interaction terms between the connection terms (i.e. the metric) and the neural fields are given in Eq. (37).

$$\begin \mathcal (\phi ,\phi ^,D_ \phi ,D_\phi ^,R)&= \sqrt\left( g^D_ \phi ^ D_ \phi \right. \nonumber \\ &\left. + m^|\phi |^ + R\right) \end$$

(37)

This equation is also known as the Hilbert action (or the Einstein-Hilbert action) and will in 3 spatial dimensions (with 1 time dimension) give the field equations for general relativity. In 2 spatial dimensions the interaction between the neural and connectivity field will cause non-trivial dynamics. Using the dynamical equation derived from Eq. (36) we will show that the proposed theory inherently incorporates Hebbian and non-Linear Hebbian learning (Section 3).

2.4 Multi-layer dynamics

We now expand our study of the single (bi-) layer with several interacting layers. To obtain a theory which allows for analytical study we can perturb the connectivity around a state with only self-connections for each bi-layer and show that the perturbation of this (trivial) connectivity will give non-trivial dynamics. The connection between neural units gives the interaction between the different cortical layers and is a matrix varying over the cortical surface (n \(\times \) n matrix for n-cortical layers) and for non-interacting layers it reduces to the identity matrix. We can modulate this matrix using a unitary matrix, \(\textbf\), using the fact that the Lagrangian (Eq. (10) is invariant to unitary transformations.

$$\begin \begin \iint _U \phi&\rightarrow \iint _\textbf^(r_0) \textbf(r)U\phi \end \end$$

(38)

Note that the perturbation is 0 at \(r_0\) but varies with r. The dynamical equation will be given by a generalisation of Eq. (8), where we integrate over retarded time.

$$\begin \begin \mathbf &= \iint _U\textbf^(r_0) \textbf(r)\vec \end \end$$

(39)

Expanding Eq. (39) and using the same steps as in Section 2.2 we get the dynamical Eq. (40).

$$\begin \begin 0&= g^\partial _\partial _\left( \textbf\vec \right) -m^2\textbf\vec \\ 0&= \left( g^\partial _\partial _-m^2 \right) \left( \textbf \vec \right) \\ \end \end$$

(40)

Writing out the expansion of \(\textbf\) and the derivative terms shows that Eq. (40) is not invariant to transformations but gives us instead relation 41.

$$\begin \begin 0&= \textbfg^\partial _\partial _\vec + 2g^\partial _\textbf\partial _\vec - m^2\textbf\vec \\ \end \end$$

(41)

Similar to the analysis in Section 2.2, we can introduce a connectivity field, or gauge field, to compensate for the 3 terms in Eq. (41). The covariant derivative given by Eq. (42) will compensate for the terms in relation 41.

$$\begin \begin D^&= \partial _-(\textbf^\partial _\textbf)\\ \end \end$$

(42)

The dynamical field is invariant to transformations given by \(\textbf\) as shown with the “commutation” relation in Eq. (43).

$$\begin \begin \left( g^\partial _\partial _-m^2 \right) \left( \textbf \vec \right)&=}\left( g^D_D_-m^2 \right) \vec \\ \end \end$$

(43)

The dynamics of the neural and connectivity field interaction is given in Eq. (44).

$$\begin \begin g^\left( D_D_+m^2\right) \vec &= 0\\ \end \end$$

(44)

The Lagrangian density is given in Eq. (45) and is invariant to unitary transformations.

$$\begin \mathcal (\vec ,\vec ^,\textbf \vec ,\textbf \vec ^) = g^D_ \vec ^ D_ \vec + m^|\phi |^ \end$$

(45)

The differential operator, \(\textbf\), can be parametrised for unitary transformations using a similar set-up that we used in Section 2.2. For a N-layered cortical surface, we have N generators \(\mathbf \) for special unitary modulations (transformations) and the covariant derivative is defined in Eq. (46).

$$\begin \begin D_&= \partial _-ig\textbf_\\ \end \end$$

(46)

Where \(\textbf\) and \(\textbf_\)are given by the following expressions (where we are summing over the repeated a index).

$$\begin \begin \textbf&= e^^a\theta ^a}\\ g\textbf_&= \partial _\theta ^a\textbf^a\\ \end \end$$

(47)

The connectivity field is defined to be that of \(\textbf\). The above expression states explicitly that \(\textbf_\) is derived from a gradient field. However, like the analysis of single bi-layer dynamics, we can introduce solenoids fields in \(\textbf_\) by expanding the variation of the connectivity field. We will again parameterise the variation using a complex variable and its conjugate, instead of a real parameter as in Eq. (47) (i.e. \(\theta \in \mathbb \cong \mathbb ^2\)). The Lagrangian density does not allow for self interaction of the connectivity field and the simplest term that can be added to introduce a self interaction of the connectivity field is the minimal coupling for the connectivity field. This minimal coupling is given by the curvature of the connection field \(\textbf_\) (Eq. (48)).

$$\begin \begin \textbf_^&= \partial _A_^-\partial _A_^+ g\sum }A_^A_^\\ \end \end$$

(48)

The term pre-multiplying the connectivity field, \(f^\), is determined by the type of transformations that the connectivity field generates, in this case it is the structure constants of the Lie Algebra SU(n). The full Lagrangian density will be given by Eq. (49).

$$\begin \begin \mathcal &= D^ \vec D_ \vec + m^\vec \vec -\frac\textbf_^\textbf^\\ \end \end$$

(49)

These are highly non-linear equations but are the least complex that ensure unitary invariance of the dynamics.

In summary, the above Eq. (