On the stability of equilibria of the physiologically-informed dynamic causal model
aa r X i v : . [ q - b i o . N C ] J a n On the stability of equilibria of thephysiologically-informed dynamic causal model
Sayan Nag [email protected]
January 15, 2021
Abstract
Experimental manipulations perturb the neuronal activity. This phe-nomenon is manifested in the fMRI response. Dynamic causal model andits variants can model these neuronal responses along with the BOLD re-sponses [1, 2, 3, 4, 5] . Physiologically-informed DCM (P-DCM) [5] givesstate-of-the-art results in this aspect. But, P-DCM has more parameterscompared to the standard DCM model and the stability of this particularmodel is still unexplored. In this work, we will try to explore the stabilityof the P-DCM model and find the ranges of the model parameters whichmake it stable.
Experimental perturbations lead to the fluctuations in the neuronal activitywhich are known as neuronal responses. These neuronal responses drive the va-soactive signals which in turn drive the blood flow responses. The changes in theblood flow responses lead to changes in the blood volume and deoxyhemoglobincontent which together give rise to BOLD signals.To model this entire flow which occurred in the brain due to some activ-ity and find the causal interactions between the different inter-connected brainregions biophysical models like Dynamic Causal Models (DCMs) have been gen-erated. DCM explains the causality between different regions. The connectionbetween different regions are given in the form of a probabilistic graphical modelwhere the nodes represent the brain regions. The effective connectivity betweenthese regions are parameterized in terms of the coupling parameters. In otherwords, the effective connectivity between two regions or nodes is expressed interms of the conditional dependency between those two regions.The physiologically-informed DCM (P-DCM) is a physiologically realisticstate-of-the-art DCM variant which gives better model evidence when comparedto other DCM approaches. It introduces a factor in the connection between theexcitatory and inhibitory neuronal population. Along with this, there are somemodifications in the neurovascular coupling where a feed-forward model has1een introduced instead of feedback-based model. It also introduces a dynamicviscoelastic effect in the balloon model. Altogether, it introduces more parame-ters compared to the vanilla DCM, renders dynamic stability to the system andmakes the model more realistic from physiological point of view.Equilibrium points are not always stable. It is important to analyze thestability of the equilibrium points they play different roles in determining thedynamics of a system. In this work, considering each of the model equations wewill look into the parameters and their corresponding ranges which make thepDCM model dynamically stable near respective equilibrium points.
The P-DCM model consists of 3 sub-models, namely, the neuronal model, theneurovascular coupling, the hemodynamic and balloon models and the BOLDmodel. The BOLD model consists a set of equations which is empirically deter-mined. We will focus on the neuronal model, the neurovascular coupling andthe hemodynamic and balloon models which were the main contributions of theP-DCM paper [5].
The neuronal model consists of the excitatory and inhibitory states. The dy-namics are dictated by the following state-space differential equations: dx E ( t ) dt = − σx E ( t ) − µx I ( t ) + cu ( t ) (1) dx I ( t ) dt = λx E ( t ) − λx I ( t ) (2)Here, σ represents the self-intrinsic connectivity of the excitatory neuron. λ isthe inhibitory gain factor. The excitatory-inhibitory mutual connection is givenby µ . Neurovascular coupling refers to the changes in CBF and CBV due to changein the neuronal activity. A feedforward based NVC model has been chosen inP-DCM which is given by the following set of differential equations: da ( t ) dt = − ρa ( t ) + x E ( t ) (3) df ( t ) dt = φa ( t ) − χ ( f ( t ) − (4)Here, a(t) is the vasoactive signal. Using the above equations, it transformsthe neuronal response x E ( t ) to the blood flow response which is given by f(t), ρ ,2 , χ respectively represents the decay of vasoactive signal, the gain of vasoactivesignal and the decay of blood inflow signal. The hemodynamic model dictates the dynamics of the hemodynamic variablescalled blood volume v(t) and the deoxyhemoglobin content q(t) with respect tothe blood inflow f(t) and the blood outflow f out ( v ) . dv ( t ) dt = 1 t MT T [ f ( t ) − f out ( v, t )] (5) df ( t ) dt = 1 t MT T [ f ( t ) E ( f ) E − f out ( v, t ) q ( t ) v ( t ) ] (6)Here, t MT T is the mean transit time the blood takes to pass the veins. E(f)is the oxygen extraction fraction when the blood inflow is f(t) and E is the netoxygen extraction at rest. E(f) is given as: E ( f ) = 1 − (1 − E ) /f (7)Considering the visco-elastic effect the balloon model is given as: f out ( v, t ) = 1 τ + t MT T ( t MT T v ( t ) /α + τ f ( t )) (8)Here, τ is the visco-elastic time constant and α is the Grubb’s exponent. Linear Stability Analysis refers to the stability analysis of a system after thelinearization of the differential equations at respective equilibrium points aredone. Equilibrium refers to the state of a system which does not change. Equi-librium point(s) can be estimated by setting the derivative(s) of the differentialequation(s) (describing the system) to zero.A Jacobian matrix is formed by the first order partial derivatives of a sys-tem of differential equations. The eigen values of a Jacobian matrix can berepresented in the form of complex numbers. The complex part of the eigen-value contributes an oscillatory component, but, the real part determines thestability of the equilibrium point.An equilibrium point of the differential equation defining a system is consid-ered to be: (a) stable if all the eigenvalues of the Jacobian matrix evaluated atthat equilibrium point have negative real parts and (b) unstable if at least oneof the eigenvalues of the Jacobian matrix evaluated at that equilibrium pointhas a positive real part.In this section we will show the detailed analysis of P-DCM model, especiallywe will try to investigate the values of the model parameters which make themodel dynamically stable near the respective equilibrium points.3 .1 Neuronal model
Equations (1) and (2) give the neuronal dynamics of the P-DCM Model. Now,making both of them equal to 0, we will get the equilibrium values of the system.The equilibrium values of x E and x I are the same and equals to cu/ ( σ + µ ) . Now,we will try to find the nature of stability of the equilibrium point by computingthe Jacobian matrix. The Jacobian matrix is represented as: J NeuronalModel = (cid:20) − σ − µλ − λ (cid:21) (9)The characteristic equation in terms of eigen values (s) is given as: s + ( σ + λ ) s + ( µλ + σλ ) = 0 (10) s and s are the roots of the above equation given as: s = − ( σ + λ ) + p ( σ + λ ) − µλ + σλ ) (11) s = − ( σ + λ ) − p ( σ + λ ) − µλ + σλ ) (12)So, for the equilibrium point to be a stable one (and hence the system to bestable), 1. σ + λ has to be more than 0. 2. p ( σ + λ ) − µλ + σλ ) can be ≤ ,3. σ + λ > mod p ( σ + λ ) − µλ + σλ ) , if p ( σ + λ ) − µλ + σλ ) > By definition each of σ and λ is positive. So the first condition is satisfied.Considering λ > 0,for the third condition, we see that µ + σ has to be greaterthan 0. By, definition µ is also positive, so third condition holds true. For thesecond condition we get: ( σ − λ ) ≤ µλλ − p µλ ≤ σ ≤ λ + 2 p µλ The given ranges for µ and λ in the P-DCM paper [5] are: < µ < . , < λ < . Now, using these ranges for µ and λ in the above expression showing thepermissible range of σ and also considering σ to be positive (by definition andalso to ensure stability), we find: < σ < . In the P-DCM paper, σ was made to vary within 0 and 1.5. So as alreadycomputed above, the mentioned range of sigma (though it can be extended to1.64) makes the equilibrium a stable focus and thus makes the system asymp-totically stable. 4 .2 Neurovascular Coupling (NVC) Equations (3) and (4) give the NVC dynamics of the P-DCM Model. Now,making both of them equal to 0, we will get the equilibrium values of the system.The equilibrium values of a and f are respectively x ∗ E ρ and φx ∗ E ρχ + 1 . Here, x ∗ E isthe steady-state value of x E . Now, we will try to find the nature of stability ofthe equilibrium point by computing the Jacobian matrix. The Jacobian matrixis represented as: J NV C = (cid:20) − ρ φ − χ (cid:21) (13)The characteristic equation in terms of eigen values (s) is given as: ( s + χ )( s + ρ ) = 0 (14) s and s are the roots of the above equation given as: s = − χ (15) s = − ρ (16)So, for the equilibrium point to be a stable one (and hence the system to bestable), both χ and ρ have to be positive. Based on the values mentioned in theP-DCM paper ( χ = 0 . and ρ = 0 . ), the equilibrium is a stable focus. Putting the values of E ( f ) and f out ( v, t ) in equations (5) and (6) and makingboth of them equal to 0, we will get the equilibrium values of the variables ofthe system. The equilibrium values of v and q are respectively v eq and q eq . v eq = ( f ∗ ) α (17) q eq = v eq f ∗ E ∗ f ∗ out E (18)Here, f ∗ , f ∗ out and E ∗ are respectively the steady-state values of f , f out and E . Now, we will try to find the nature of stability of the equilibrium point bycomputing the Jacobian matrix. The Jacobian matrix is represented as: J H = − v /α − eq α ( t MTT + τ ) α − q eq v (1 /α − eq t MTT + τ + τq eq v − eq t MTT − ( v (1 /α − eq t MTT + τ + τf ∗ t MTT v eq ) (19)So, the eigen values s and s are given as: s = − v /α − eq α ( t MT T + τ ) (20)5 = − ( v (1 /α − eq t MT T + τ + τ f ∗ t MT T v eq ) (21)So, for the equilibrium point to be a stable one (and hence the system to bestable), both s and s have to be negative. Based on the values mentioned inthe P-DCM paper ( < t MT T < and < τ < ), the equilibrium is a stablefocus. Linear stability analysis near the equilibrium points for the system of differentialequations defining the P-DCM model gives an idea about the ranges of theparameters which make the model dynamically stable. It has been found thatthe ranges of the model parameters defined in the actual P-DCM paper [5]confer dynamical stability on the equilibrium points of the model differentialequations.
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