On propagation in networks, promising models beyond network diffusion to describe degenerative brain diseases and traumatic brain injuries

Introduction: Connections among neurons form one of the most amazing and effective network in nature. At higher level, also the functional structures of the brain is organized as a network. It is therefore natural to use modern techniques of network analysis to describe the structures of networks in the brain. Many studies have been conducted in this area, showing that the structure of the neuronal network is complex, with a small-world topology, modularity and the presence of hubs. Other studies have been conducted to investigate the dynamical processes occurring in brain networks, analyzing local and large-scale network dynamics. Recently, network diffusion dynamics have been proposed as a model for the progression of brain degenerative diseases and for traumatic brain injuries. Methods: In this paper, the dynamics of network diffusion is re-examined and reaction-diffusion models on networks is introduced in order to better describe the degenerative dynamics in the brain. Results: Numerical simulations of the dynamics of injuries in the brain connectome are presented. Different choices of reaction term and initial condition provide very different phenomenologies, showing how network propagation models are highly flexible. Discussion: The uniqueness of this research lies in the fact that it is the first time that reaction-diffusion dynamics have been applied to the connectome to model the evolution of neurodegenerative diseases or traumatic brain injury. In addition, the generality of these models allows the introduction of non-constant diffusion and different reaction terms with non-constant parameters, allowing a more precise definition of the pathology to be studied.


REACTION DYNAMICS
In this section, we consider in details the reaction terms considered in the main document, i.e., the logistic growth term the neutral Allee effecct and the strong Allee effect The evolution dynamics of the concentration of a given substance, θ, that undergoes a reaction equation, is given by the following differential equation where θ 0 is the initial concentration and r is the reaction rate.In the case of the exponential growth of a population due to a linear growth rate, θ = rθ, r represents the constant per capita growth rate, θ/θ, that is independent of the population size.Exponential growth is possible in nature only for short periods of time, after which population (or concentration) saturation occurs.Such a behaviour is modelled by the logistic growth (S1).
The differential equation (S4) can be analytically solved for each of the reaction terms presented, (S1), (S2) and (S3).Just as an example we present the solution in the case of the logistic term (S1).
(S5) However, in this section, we want to qualitatively study the different dynamical behaviour given by Eq. (S4) occurring for the reaction terms here discussed and for different initial conditions.Fig. 1a shows the logistic function (Eq.(S1), Eq. ( 8) in the main document), Fig. 1d the per capita growth rate, i.e. θ/θ, and Fig. 1g the evolution dynamics of Eq. (S4) starting from different initial conditions, θ 0 ., per capita growth rate, θ/(θ) and evolution dynamics (S4) for the reaction terms used in this section.In all the figures the concentration threshold is fixed θ c = 0.1.As for the evolution dynamics, we use different initial condition, θ(0) = θ 0 , reported in the figures' labels.
It is important to emphasize the profoundly different behavior of the dynamics determined by the different reaction terms.The simple logistic term induces an initial exponential growth in the concentration leading to saturation at 1, regardless of the initial condition, that can be seen also in the analytical solution (S5).The neutral Allee effect show similar behaviour if the initial condition is above the threshold, θ 0 > θ c , while for θ 0 ≤ θ c there is no dynamics since θ ′ (t) = 0 and θ(t) = θ 0 .The strong Allee effect is the reaction term with the most peculiar dynamics, since the saturation time at 1 is much slower than in the other two cases, and, for initial conditions below θ c , the per capita growth rate is negative and the concentration goes to zero.
Obviously, many different reaction terms can be introduced, capable of describing very different phenomenology, which may be useful for modelling different dynamics occurring in traumatic brain injuries or in progressive brain diseases.

REACTION-DIFFUSION DYNAMICS
In this section we discuss some known results about reaction diffusion process in one dimension and we compare the reaction diffusion dynamics with the simple diffusion dynamics.
The prototypical model for reaction-diffusion in one dimension is the non linear diffusion equation where the non-linear function, g(θ), could be one of the functions discussed in the above section.
In the case of the logistic non-linear term (S1), the reaction-diffusion equation is the one originally proposed in the seminal contribution by R. A. Fisher (Fisher, 1937) and A.N. Kolmogorov together with I.G.Petrovskii and N.S.Piskunov (Kolmogorov et al., 1937), which is commonly called FKPP equation.In the case of bounded initially conditions, i.e., when θ(x, 0) is different from zero only in a bounded region, the FKPP equation shows travelling wave dynamics, i.e. θ(x, t) = Θ(x − vt), that is a travelling wave of shape Θ(z) with speed In the case of different reaction term, it is possible to determine upper and lower bound for the front speed (Aronson and Weinberger, 1978) Reaction terms for which g ′ (0) = sup θ g(θ)/θ, i.e., the maximum per capita growth rate is in the origin (see Fig. s 1a and 1d), are called FKPP-like, and their front speed is that of the FKPP equation ((S7)).When the maximum per capita growth rate is not in zero (see Fig. s 1b and 1e) the dynamics are ruled by the region of θ(x, t) that are near the maximum per capita the growth rate.
The asymptotic evolution of the total quantity of substance (see Eq. ( 12) of the main document) in the case of a propagating front, is given by Examples of propagation dynamics in the case of FKPP-like dynamics (front propagation and total quantity of concentration in Fig. s 2a and 2b, respectively) and not FKPP-like dynamics (front propagation and total quantity of concentration in Fig. s 2a and 2b), can be found in Fig. 2, together with the case of a simple diffusion dynamics.The comparison is very clear, a simple diffusion dynamic does not allow an increase in concentration and does not have true propagation, but simply a spreading of concentration.Propagation dynamics, on the other hand, are characterised by a moving front with a speed determined by the parameters of the system.
In the case of neutral or strong Allee effect, when the initial condition is below the concentration threeshold, or when diffusion time is faster than reaction time, it could be happen, also in the presence of a reaction diffusion dynamics, that front propagation may not take hold, and in this case we talk about quenching of the dynamics (see for example

NUMERICAL INTEGRATION
The diffusion equation on graph has an exact solution that is almost never easy to calculate, so numerical computation methods are used.There are many software packages that calculate diffusion on graph.In the case where a reaction term is added, the exact analytical solution, in general, does not exist and one must necessarily resort to numerical computation methods, which, however, as far as we know, are not present in any software package.In this section we will present a simple numerical integration method that is based on the discretization of the time of the evolution dynamics.
Introducing a discrete integration time, ∆t, the discrete-time version of the diffusion equation (S11) can be already found in Eq. (2) of the main paper, that we rearrange here as In order to introduce the reaction term in the numerics, we consider a discrete solver, for time step ∆t, of the simple reaction equation θ ′ = rg(θ) as Composing (S12) with (S13) we obtain an approximate solution in two steps for Eq. ( 7) of the main paper To guarantee the stability of the numerical solution are sufficient to fulfill the following conditions In the case of neutral Allee effect (S2), the function |g ′ (θ)| in θ = θ c is not defined.One must use |g ′ + (θ c )| as the right derivative of g ′ (θ) in θ c .

Figure 1d .
Figure 1d.Per capita growth rate for the logistic reaction term.

Figure 1e .
Figure 1e.Per capita growth rate for the neutral Allee effect.

Figure 1f .
Figure 1f.Per capita growth rate for the strong Allee effect.

Figure 1g .
Figure 1g.Evolution dynamics for logistic reaction term for different initial conditions.

Figure 1h .
Figure 1h.Evolution dynamics for neutral Allee effect for different initial conditions.

Figure 1i .
Figure 1i.Evolution dynamics for strong Allee effect for different initial conditions.

Figure 1 .
Figure1.Reaction terms, g(θ), per capita growth rate, θ/(θ) and evolution dynamics (S4) for the reaction terms used in this section.In all the figures the concentration threshold is fixed θ c = 0.1.As for the evolution dynamics, we use different initial condition, θ(0) = θ 0 , reported in the figures' labels.
Fig.s 1d and 1g of the main paper).

Figure 2b .
Figure 2b.Growth of concentration with logistic reaction dynamics.

Figure 2d .
Figure 2d.Growth of concentration with neutral Allee effect.

Figure 2f .
Figure 2f.Stationarity of concentration with diffusion dynamics.

Figure 2 .
Figure 2. Comparison of diffusion dynamics and reaction diffusion propagation dynamics.The initial condition is localized in x = 200 with a total concentration M (0) = 1.