text:
# Conduction in the Heart Wall: Helicoidal Fibers Minimize Diffusion Bias

## Abstract

The mammalian heart must function as an efficient pump while simultaneously conducting electrical signals to drive the contraction process. In the ventricles, electrical activation begins at the insertion points of the Purkinje network in the endocardium. How does the diffusion component of the subsequent excitation wave propagate from the endocardium in a healthy heart wall without creating directional biases? We show that this is a consequence of the particular geometric organization of myocytes in the heart wall. Using a generalized helicoid to model fiber orientation, we treat the myocardium as a curved space via Riemannian geometry, and then use stochastic calculus to model local signal diffusion. Our analysis shows that the helicoidal arrangement of myocytes minimizes the directional biases that could lead to aberrant propagation, thereby explaining how electrophysiological principles are consistent with local measurements of cardiac fiber geometry. We discuss our results in the context of the need to balance electrical and mechanical requirements for heart function.

## Introduction

The heart can be viewed as a mechanical device or as an electrical device, and these must work in concert for billions of pumping contractions over an average lifetime. During every beat, the cardiac fiber geometry must support safe propagation of the electrophysiological contraction signal, or irregular patterns in the conduction wave could lead to potentially deadly arrhythmias1. In the ventricular system, the earliest activation sites are associated with the entry points of the Purkinje network in the sub-endocardium2. In the present article, our aim is to study how the excitation wave propagates from these sites without introducing directional biases in its diffusion component. The answer, we shall show, lies in the helicoidal arrangement of heart wall myofibers.

The heart is largely composed of locally parallel bundles of contractile cardiac muscle cells embedded in an extracellular matrix3. Analysis methods from differential geometry have shown that the orientations of these fibers approximate a minimal surface in $${{\mathbb{R}}}^{4}$$, the generalized helicoid model (GHM)4. Such helicoidal fiber geometry affords great mechanical strength5, as the club-like appendages of the mantis shrimp6 and the tough dermal armor of certain fish7 illustrate; indeed, the enhanced damage tolerance is leading to applications in material science as well8,9.

The organization of the cardiomyocytes (i.e. their local orientation) determines a number of essential properties of the heart, including the diffusive propagation of the contraction signal and mechanical efficiency10. Several studies have postulated mechanistic reasons for the local helicoidal geometry (e.g.11,12) and, from an evolutionary perspective, it seems to emerge with the need for high blood pressure in mammals and birds13. Mechanical considerations alone do not suffice, however; cardiac tissue is unique in that mechanical resilience must coexist with active contraction and signal propagation.

Electrophysiologically, the cardiomyocyte fibers exert a powerful local effect on the rate of signal propagation: diffusion within fibers (and between them via end-to-end connections), is much faster than in directions orthogonal to the fibers14,15,16. Based on this, heart tissue has been modeled as a Riemannian manifold with the metric tensor determined by the local fiber direction. An important result17, which we confirm, is that the tissue can be approximated by a manifold with negative scalar curvature. Mathematically, this enhances diffusion in abstract manifolds18,19 and in the walls of the ventricles17; it also plays a role in models of molecular diffusion20,21.

In more detail, the effect of the cardiac fiber architecture on diffusion of the contraction signal has important physiological and biomedical implications. Disturbances in the signal propagation can lead to ventricular fibrillation and death, and the transmural rotation of the fibers can lead to wave breakup and chaotic behavior of wave filaments. Taken together, this implies that the fiber geometry could be a potentially dangerous source of destabilization22,23,24,25,26. How the heart wall fiber geometry balances the competing mechanical need for efficient contraction, with the electrophysiological requirement that irregular wave patterns be avoided, is an open question.

Classically, the propagation of the excitation signal is described by a reaction-diffusion equation, where the diffusion component is anisotropic and spatially varying27,28. Because we are interested in the local behavior around the insertion points of the Purkinje network in the endocardium, we focus on the diffusion term and introduce the effect of local fiber geometry via a GHM (see Fig. 1). The duality between the diffusion equation and Brownian motion (BM) allows us to move beyond the average ensemble behavior to individual sample functions and their variance (see Fig. 2). We analyze the properties of BM as an Ito process on the cardiac manifold, and our key results are that (i) there is an acceleration effect on the diffusion process due to the GHM structure, (ii) the GHM structure minimizes local directional biases in the diffusion component of the conduction wave, and (iii) within the heart wall tangent plane, the stochastic component of diffusion, i.e., its variance, becomes isotropic very quickly. All three results require that there be transmural rotation of fiber orientation in the heart wall. In the end, then, the helicoidal fiber geometry supports efficient mechanical operation11,29 while also reducing directional biases in electrical signal propagation.

## Model

### The Generalized Helicoid Model

To first approximation, the fiber geometry in the left ventricle has long been empirically known to show a transmural rotation through the wall, essentially irrespective of position30,31. This forms the basis of rule-based models of the ventricular fiber structure as well32. To model this, one can define the z-axis to lie along the transmural direction, and define an orientation (or fiber angle) function θ(x, y, z), which describes the angle of rotation of the local myofiber at the given position.

The full expression for the orientation function of the GHM may be written as4:

$$\theta (x,y,z)=\arctan (\frac{{k}_{T}x+{k}_{N}y}{1+{k}_{N}x-{k}_{T}y})+{k}_{B}z$$
(1)

where k B , k T , $${k}_{N}\in {\mathbb{R}}$$ are constants (see Fig. 3 for an illustration). The GHM is a local model, which assumes the fibers have no component out of the local tangent plane to the wall (i.e. no imbrication angle). It describes a series of “planes” of fibers along the transmural axis, corresponding to the local heart wall. Thus, the orientation function above allows defining the GHM as a vector field33 $$v\,:{{\mathbb{R}}}^{3}\to {{\mathbb{R}}}^{3}$$:

$$v(x,y,z)=(\cos (\theta (x,y,z)),\,\sin (\theta (x,y,z)),0)$$
(2)

which is illustrated for several parameter values in Fig. 4 using 3D streamlines.

Recent works using diffusion imaging suggest that |k N | and |k T | are quite small across species, whereas k B is always distinctly non-zero4,33. In the present article, therefore, we also analyze a GHM where the in-plane curvatures vanish (i.e. k T  = k N  = 0).

### Stochastic Calculus on the Cardiac Manifold

Previous work by Young and Panfilov17 has modeled diffusion propagation of the electrophysiological signal responsible for cardiomyocyte contraction using Riemannian geometry. The essence of this approach is to consider distance measures in the heart wall to be warped by the local anisotropy of the fibers.

Let $$({{\mathbb{R}}}^{3},g)$$ be a Riemannian manifold. Consider the following moving frame field to be a basis for the tangent space:

$$[\begin{array}{c}{\hat{e}}_{1}\\ {\hat{e}}_{2}\\ {\hat{e}}_{3}\end{array}]=[\begin{array}{ccc}\cos (\theta ) & \sin (\theta ) & 0\\ -\sin (\theta ) & \cos (\theta ) & 0\\ 0 & 0 & 1\end{array}]$$
(3)

Then, in this dynamic frame, the metric tensor is given by:

$$\tilde{g}={\rm{diag}}([{v}_{f}^{-2},{v}_{t}^{-2},{v}_{t}^{-2}])$$
(4)

which is the inverse of the diffusivity tensor $$\tilde{D}$$ in local coordinates. In keeping our notation consistent with prior work17, we denote the diffusivity entries along the local fiber direction and in the plane orthogonal to the local fiber as $${v}_{f}^{2}$$ and $${v}_{t}^{2}$$, since they are proportional to the squares of the conduction velocities (e.g.34). We further note that previous work utilized an orthotropic metric tensor, with a third diffusion speed constant accounting for the propagation rate within the laminar sheets of the heart17. Here, we have simplified our analysis by considering only one speed in directions transverse to the fiber. Empirical measurements35 showing that the difference between v f and transverse propagation speeds were much larger than the difference in speeds within the transverse plane suggest that this simplification is reasonable.

Intuitively, the metric tensor describes infinitesimal lengths and distances on a manifold. In this case, g describes the curved space in which the signal moves, where the warping is caused by the presence of the fibers. In other words, instead of explicitly modeling the fibers, we model space itself as being intrinsically curved, such that the same movement in different directions leads to different distances being covered (with movement faster along fibers than orthogonal to them).

On the cardiac manifold, it is possible to describe diffusion as a continuous space-time stochastic process using the Ito calculus, which generalizes classical calculus to handle random variables. This is in contrast to the standard method of analyzing diffusion in the heart via partial differential equations (PDEs), as it allows probabilistic interpretations of propagation behavior that deterministic approaches do not. Informally, the PDEs approach describes the macroscopic behavior of diffusion, while the stochastic representation describes the nature of a single infinitesimal member of the ensemble that combines to generate the macroscopic average behavior. Another advantage is that the stochastic description of diffusion can permit intuitive and analytic understanding, whereas the PDEs model is generally analytically intractable.

First, the Laplace-Beltrami operator on a Riemannian manifold (the analogue of the Laplacian in Euclidean space) can be defined in local coordinates by

$${{\rm{\Delta }}}_{g}f=\frac{1}{\sqrt{|{\rm{\det }}\,g|}}\frac{\partial }{\partial {x}^{j}}(\sqrt{|{\rm{\det }}\,g|}\,{g}^{ij}\frac{\partial f}{\partial {x}^{i}})$$
(5)

where gij are elements of the inverse metric tensor g−1. Note that the Einstein summation convention is used to sum over repeated indices. Then, diffusion on a Riemannian manifold is a stochastic process with an infinitesimal generator given by $${\mathscr{A}}={{\rm{\Delta }}}_{g}/2$$36,37. Further, define the drift coefficient $${b}^{i}={g}^{jk}{{\rm{\Gamma }}}_{jk}^{i}$$ and diffusion coefficient $${\rm{\sigma }}=\sqrt{{g}^{-1}}$$, where $${{\rm{\Gamma }}}_{jk}^{i}$$ are the Christoffel symbols. Then the following system of stochastic differential equations (SDEs) describes a diffusion process on the manifold37,38:

$$d{X}_{t}^{i}={{\rm{\sigma }}}_{j}^{i}({X}_{t})\,d{B}_{t}^{j}-\frac{1}{2}{b}^{i}({X}_{t})\,dt$$
(6)

where i {1, 2, 3} and B t is a three-dimensional Wiener process, i.e. $${B}_{t}^{i}\sim {\mathscr{N}}\mathrm{(0},t)$$. The Ito process defined by this system of SDEs describes the behavior of diffusion on the cardiac manifold.

### Diffusion on the Cardiac Manifold

Classically, the propagation of the contraction signal in the heart can be modeled as a reaction-diffusion equation, with a spatially varying diffusion tensor D (with components Dij) depending on the local fiber orientation27,28, via

$$\frac{\partial u}{\partial t}=\frac{\partial }{\partial {x}_{i}}{D}_{ij}\frac{\partial u}{\partial {x}_{j}}+{\rm{\Phi }}(u,\overrightarrow{v});\,\frac{\partial \overrightarrow{v}}{\partial t}=\overrightarrow{{\rm{\Psi }}}(u,\overrightarrow{v})$$
(7)

where $$\overrightarrow{{\rm{\Psi }}}$$, Φ are non-linear reaction functions describing the cardiomyocyte electrophysiology, while u and $$\overrightarrow{v}$$ track cellular activation state variables, and the first term is generated by the standard anisotropic diffusion generator $$\tilde{ {\mathcal L} }=\frac{\partial }{\partial {x}_{i}}({D}^{ij}\frac{\partial }{\partial {x}_{j}})$$. Such studies tend to convert between the diffusion PDE and Riemannian geometric representations by defining the metric to be the inverse diffusivity tensor, with components D ij . However, from a stochastic perspective, it can be shown that a more natural approach is to consider the metric tensor to be the adjugate matrix of the diffusion tensor39,40, as it expresses isotropic diffusion in the curved space as anisotropic diffusion in Euclidean space.

There is a deep duality between the heat (isotropic diffusion) equation and BM on the cardiac manifold: the heat kernel, i.e. the fundamental solution to the heat equation $$(\frac{\partial }{\partial t}-{\mathscr{A}})u=0$$, is exactly the transition density function of BM on the manifold38,41. More intuitively, we can formalize the notion that the heat equation describes the expected behavior of an ensemble of BM processes by noting that $$u(t,x)={\mathbb{E}}[f({X}_{t,x})]$$ solves the manifold heat equation with initial conditions u(0, x) = f(x), where Xt,x is a BM on the manifold starting from x37,42. Thus, analysis of the Ito diffusion process above corresponds to understanding the heat equation on the manifold, which comprises the diffusion term in the reaction-diffusion equation above. Figure 5 illustrates how the curved GHM space affects diffusion from a point in a helicoidal medium, such as from an excitation site of the Purkinje network in the endocardium2.

Herein, we focus on the behavior of diffusion on the manifold from a stochastic perspective, to understand how the GHM affects contraction signal propagation.

## Results

### The GHM Manifold Possesses Negative Ricci Curvature

As noted above, the cardiac tissue can be considered a Riemannian manifold with a metric tensor defining spatial distances on the manifold g, warped by the presence of the fibers. For any such manifold, the curvature of the space can be measured via the Ricci curvature scalar R, which in the case of the GHM is given by (see Supplemental Information (Derivations): Ricci Curvature of the Cardiac Manifold)

$$R=\frac{-{k}_{B}^{2}{({v}_{f}^{2}-{v}_{t}^{2})}^{2}}{2{v}_{f}^{2}},$$
(8)

in agreement with previous calculations by Young and Panfilov17. Since the scalar curvature is spatially constant and always negative, it accelerates diffusion propagation on the manifold.

As also noted in previous work17,43, this acceleration can be seen in the fact that, for small t, the volume of activated tissue grows as

$${\rm{Vol}}({B}_{t})=\frac{4\pi }{3}{v}_{f}{v}_{t}^{2}{t}^{3}(1-\frac{R}{30}{t}^{2}+O({t}^{4}))$$
(9)

where B t is a metric ball at time t. We note another connection to the speed of diffusion lies in considering the short-time asymptotic expansion of the heat kernel p(t, q1, q2), the fundamental solution of the diffusion equation $$(\frac{\partial }{\partial t}-{\mathscr{A}})u=0$$ on the manifold, where $$t\in {\mathbb{R}}$$ and $${q}_{1},{q}_{2}\in {{\mathbb{R}}}^{3}$$. For the GHM manifold, it can be shown that the autodiffusion function h t (q) = p(t, q, q) can be expanded as42,44

$${h}_{t}(q)=\frac{1}{{(2\pi t)}^{3/2}}(1+\frac{R}{12}t+O({t}^{2}))$$
(10)

when t is small, showing that the local Ricci scalar curvature dominates the local behavior of diffusion on such time scales. Recalling that p is the transition density of the Ito diffusion defining BM on the manifold, we again see the duality between the stochastic and PDE formulations of the signal diffusing. One can thus interpret h t (q) as either the flow of a quantity from a point to itself or the probability of BM staying at q over time. Informally, if the probability of staying near a start point q (i.e. h t (q)) is lower, then the diffusion is flowing away from its start point q more quickly. As such, in the case that R < 0, notice that the curvature term decreases the value of h t (q), meaning that the term is encouraging the diffusion flow to move faster from its start point. Hence, negative R relates directly to faster diffusion from a point.

A natural question is whether the GHM optimizes the Ricci curvature in the variational sense, which would imply a maximization or minimization of the curvature-derived acceleration of the diffusion rate. Using the calculus of variations, we treat the Ricci curvature as a functional on the space of orientation functions. Akin to classical calculus, the first variation shows whether an input is an extremal point in function space, while the second variation classifies its type. We consider two different function spaces: the set of “planar” fiber fields (with no imbrication angle) and the space of general 3D unit vector fields.

Using variational techniques, it can be shown that the GHM lies at a stationary point of the Ricci curvature scalar, in the planar case (see Supplemental Information (Derivations): Variational Analysis of the GHM Ricci Curvature). In contrast, in the general space, only the GHM with k N  = k T  = 0 has vanishing first variation; the full GHM is not a stationary point for other parameter values. However, the stationary point is neither a minimum nor a maximum, as it fails to satisfy the Legendre condition, which is a necessary requirement for the function to lie at an extremum45.

We note that the variational status of the GHM as a stationary point with respect to the Ricci curvature in the full space requires the vanishing of the in-plane curvatures (k N and k T ). Recent imaging studies with GHM fits confirm that these curvatures are small in magnitude4,33. Another example of the importance of the vanishing of these curvatures will be shown in the next section, with respect to the drift vector of diffusion on the manifold.

In summary, corroborating previous analyses17, the GHM has a constant negative Ricci curvature, which has an accelerating effect on the diffusion process. Our variational analysis of the GHM shows that it always lies at a stationary point in the planar space of fiber angle functions, but it is only stationary in the full space when the in-plane curvatures (k N , k T ) vanish. It is interesting that the GHM lies only at a stationary point, rather than a minimum or maximum, perhaps indicating a balance between competing factors, such as wave speed and stability. We note that previous studies22,23,26 suggest that increased rate of transmural rotation (determined by k B ) may be linked to cardio-electrophysiological instability. Hence, although increasing |k B | would accelerate the signal propagation, this may not be beneficial to the organism overall.

### The Diffusion Drift is Minimized by the Helicoidal Architecture

The Ito diffusion process of BM on a manifold is described by a system of SDEs that is parametrized by a diffusion coefficient $${\rm{\sigma }}=\sqrt{{g}^{-1}}$$ and drift coefficient $${b}^{i}={g}^{jk}{{\rm{\Gamma }}}_{jk}^{i}$$. For the GHM, the former is given by

$$\sigma =\frac{1}{2}[\begin{array}{ccc}\varsigma +\delta \,\cos (2\theta ) & \delta \,\sin (2\theta ) & 0\\ \delta \,\sin (2\theta ) & \varsigma -\delta \,\cos (2\theta ) & 0\\ 0 & 0 & 2{v}_{t}\end{array}]$$
(11)

where ς = v f  + v t and δ = v f  − v t , while the latter can be written as

$${b}^{i}=\frac{\delta \varsigma [\,-\,{\alpha }_{i}\,\cos \,\mathrm{(2}\theta )+{\beta }_{i}\,\sin \,\mathrm{(2}\theta )]}{1+2{k}_{N}x-2{k}_{T}y+\gamma ({x}^{2}+{y}^{2})}$$
(12)

and b3 = 0, where $$\gamma ={k}_{N}^{2}+{k}_{T}^{2}$$, which intuitively measures the total local in-plane fiber curvature, α1 = −β2 = k N  + γx, and α2 = β1 = k T  − γy. Thus, to determine when the drift components vanish across the manifold, consider the Euclidean magnitude (i.e. sum of squared components) of the drift vector for the GHM:

$$\parallel b{\parallel }_{2}^{2}=\frac{\gamma {({v}_{f}^{2}-{v}_{t}^{2})}^{2}}{1+2{k}_{N}x-2{k}_{T}y+\gamma ({x}^{2}+{y}^{2})}$$
(13)

so that $$\parallel b{\parallel }_{2}^{2}=0$$ clearly implies $${k}_{N}^{2}+{k}_{T}^{2}=0$$, which holds only when k N  = k T  = 0. As such, mathematically, the drift vector is zero across the manifold if and only if k N  = k T  = 0, assuming v f  > v t . Hence, the empirical observation4,33 that k N and k T are small in nature (essentially equivalent to the assumptions of the rule-based model32) can be interpreted as a vanishing of the drift vector (i.e. deterministic bias) in the Ito process.

The calculus of variations can also provide insight into the minimization of the magnitude of the drift vector. It can be shown (see Supplemental Information (Derivations): Variational Analysis of the Ito Diffusion Drift on the GHM Manifold) that, in its most general form, the GHM does not satisfy the Euler-Lagrange equations for the Riemannian norm of the drift vector. Yet, if the in-plane curvatures vanish (i.e. k N  = k T  = 0), the GHM does lie at a minimum, since the drift is identically zero. Hence, in the space of fiber angle functions, the minimization of the diffusion drift is linked to the vanishing of the in-plane curvatures, similar to the case of the Ricci curvature.

Thus, we have shown two results linking the in-plane curvatures of the GHM to the drift vector. The first shows that the drift vector is identically zero across the cardiac manifold generated by the GHM if and only if the in-plane curvatures vanish. The second is a complementary result, showing that the full GHM (i.e. with arbitrary in-plane curvatures) is not generally a variational minimizer of the drift magnitude, unless k N and k T vanish. As such, the empirical findings4,33 that the curvature parameters k T , k N in the tangential plane to the mammalian heart wall are small relates also to the minimization of the Ito drift.

### In-Plane Variance Becomes Isotropic Exponentially Quickly

The results above suggest that several special properties of the GHM rely on the vanishing of the in-plane curvatures, k T and k N . In this section, we restrict to this case, and focus on the interpretation of the transmural rotation parameter k B .

Although there is no deterministic drift (or convection) component to the GHM encountered empirically (i.e. that with vanishing in-plane curvatures, or k N  = k T  = 0), there is still a stochastic bias in the components of the variance vector so that diffusion in some directions is favored over others. We show that for the GHM, as long as k B  ≠ 0, in the heart wall tangent (xy) plane, this bias is removed exponentially quickly. We also show that the case of k B  = 0, corresponding to no transmural rotation, is biologically problematic, because then the variance vector has a strong constant bias in the fiber direction x.

For the GHM with k N  = k T  = 0, we obtain the following system of SDEs:

$$\begin{array}{rcl}d{X}_{s}^{1} & = & \frac{1}{2}([\varsigma +\delta \,\cos (2{k}_{B}{X}_{s}^{3})]\,d{B}_{s}^{1}+\delta \,\sin (2{k}_{B}{X}_{s}^{3})\,d{B}_{s}^{2})\\ d{X}_{s}^{2} & = & \frac{1}{2}(\delta \,\sin (2{k}_{B}{X}_{s}^{3})\,d{B}_{s}^{1}+[\varsigma -\delta \,\cos (2{k}_{B}{X}_{s}^{3})]\,d{B}_{s}^{2})\\ d{X}_{s}^{3} & = & {v}_{t}\,d{B}_{s}^{3}\end{array}$$
(14)

See Fig. 2 (left) for sample trajectories obtained by numerically integrating the SDE above. One can see the influence of fibers on the diffusion processes, analogous to the acceleration along fibers seen in Fig. 1 (right).

It can be shown (see Supplemental Information (Derivations): Moments of the GHM Stochastic Diffusion Process) that the variance of the stochastic process described by these SDEs is given by:

$${\mathbb{V}}[{X}_{t}]=t\,[\begin{array}{c}\raisebox{1ex}{({v}_{f}^{2}+{v}_{t}^{2})}\!\left/ \!\raisebox{-1ex}{2}\right.\\ \raisebox{1ex}{({v}_{f}^{2}+{v}_{t}^{2})}\!\left/ \!\raisebox{-1ex}{2}\right.\\ {v}_{t}^{2}\end{array}]+\frac{\delta \varsigma [1-\exp (\,-\,ct)]}{4{k}_{B}^{2}{v}_{t}^{2}}[\begin{array}{c}1\\ -1\\ 0\end{array}]$$
(15)

where $$c=2{k}_{B}^{2}{v}_{t}^{2}$$ and that the expected value is /constant at zero. See Fig. 6 for plots of the numerical convergence of the moments of the process. Notice that the first term of the variance vector has identical components in the two in-plane directions x, y in the heart wall tangent plane, with a smaller component in the transmural direction z. The second term has components with identical magnitudes in the fiber direction x and the in-plane direction y, but with the latter having a minus sign. This causes an increase in variance in the x direction with a corresponding decrease in the y direction. Since the first term in the variance equation above has no dependence on k B , it is in fact the second term that relates to the directional bias to the variance vector caused by the transmural rotation of fibers.

Now, recall that the variance of a diffusion process can be interpreted as being related to the expected growth (movement) of the diffusion in that direction. For instance, the variance of BM in 1D isotropic Euclidean space ($${\mathbb{R}}$$) is $${\mathbb{E}}[{({B}_{t})}^{2}]=t$$; i.e. linear in time. Hence, we can gain intuition about the behavior of diffusion on the GHM by examining the asymptotics of the variance as we vary some of the parameters of the GHM diffusion model. As a simple case, suppose that v f  = v t . Then, as one would expect, $${\mathbb{V}}[{X}_{t}]=t{v}_{t}^{2}{\mathrm{[1},1,\mathrm{1]}}^{T}$$, meaning the directional anisotropy in the variance vector disappears. In particular, if v f  = v t  = 1, we get the classical result for Brownian diffusion in $${{\mathbb{R}}}^{3}$$. Next, consider the case where k B  → 0, so that there is no transmural rotation of fibers in the heart wall. It can be shown (see Supplemental Information (Derivations): Moments of the GHM Stochastic Diffusion Process) that $${\mathrm{lim}}_{{k}_{B}\to 0}\,{\mathbb{V}}[{X}_{t}]=t{[{v}_{f}^{2},{v}_{t}^{2},{v}_{t}^{2}]}^{T}$$, meaning that growth in the variance vector is accelerated in the fiber direction x, whereas it is not in y and z.

We next consider the asymptotic behavior of the terms of the variance vector in time. As t increases, the first variance term grows linearly, while the second converges exponentially fast to a constant. This can be interpreted as an isotropic linear growth of the variance vector in the heart wall tangent plane (xy) in time, with a reshaping to increase the variance of the diffusion process along fibers in the direction x while decreasing it across them in the direction y. However, the effect of this reshaping becomes quickly negligible as time increases. We also observe that this effect is identical to that of increasing k B , the helicoidal fiber rotation rate. Intuitively, a larger k B means greater homogeneity in x and y, converging to a growth averaging the diffusion rates in the fiber and non-fiber directions. Thus:

$$\mathop{{\rm{l}}{\rm{i}}{\rm{m}}}\limits_{t\to {\rm{\infty }}}\,{\mathbb{V}}[{X}_{t}]\approx t\,[\begin{array}{c}\raisebox{1ex}{({v}_{f}^{2}+{v}_{t}^{2})}\!\left/ \!\raisebox{-1ex}{2}\right.\\ \raisebox{1ex}{({v}_{f}^{2}+{v}_{t}^{2})}\!\left/ \!\raisebox{-1ex}{2}\right.\\ {v}_{t}^{2}\end{array}]=\mathop{{\rm{l}}{\rm{i}}{\rm{m}}}\limits_{{k}_{B}\to {\rm{\infty }}}\,{\mathbb{V}}[{X}_{t}]$$
(16)

as long as k B , v t , v f  ≠ 0.

Hence, in the heart wall tangent plane, the diffusion variance asymptotically expands at the average of the accelerated and non-accelerated squared rates, $$({v}_{f}^{2}+{v}_{t}^{2})/2$$; i.e. growth expands at the same rate in the x and y directions. Intuitively, the fibers create a local directional bias initially, but this vanishes exponentially fast into an isotropically accelerated propagation pattern in the heart wall tangent plane. We also showed that the bias of the variance vector grows linearly in time when k B  → 0, which is the case where all fibers lie along the x-axis (i.e. no transmural turning). There is strong empirical evidence that k B is non-zero (i.e. that there is always transmural turning of fibers)4,31,32,33. Thus, similar to how the empirical minimization of k T and k N can be interpreted as minimization of the drift vector b, the presence of non-zero k B can be viewed as reducing the bias of the variance vector within the heart wall plane.

Our expression for the variance function allows us characterize the timescale of the exponential convergence. We can consider the difference between the x and y components of the variance as the in-plane stochastic bias: $${\mathbb{V}}{[{X}_{t}]}_{x}-{\mathbb{V}}{[{X}_{t}]}_{y}=\delta \varsigma \mathrm{[1}-\exp (\,-\,ct)]/\mathrm{(2}{k}_{B}^{2}{v}_{t}^{2})$$. This difference can be seen to converge to a constant $${c}_{\infty }=\delta \varsigma /\mathrm{(2}{k}_{B}^{2}{v}_{t}^{2})$$ in time, with exponential coefficient $$c=2{k}_{B}^{2}{v}_{t}^{2}$$. Thus, the time constant of decay is given by τ = c−1. To give biophysically plausible values, we use past fits of GHM parameters in the human heart4, which show that k B  = −0.17 rad/mm, and measurements of cardiac conduction velocities, which suggest a ratio of diffusivities (i.e. ratio of squared wave velocities along vs across fibers) of approximately 4 in the ventricles46. We used a diffusivity along fibers of $${v}_{f}^{2}=3\times {10}^{-4}$$ m2/s, following a previous model34, so that $${v}_{t}^{2}={v}_{f}^{2}/4=7.5\times {10}^{-5}$$ m2/s. As a caveat, we note that measurement of the conductivities is itself a difficult problem, and there is variability in the values reported in the literature47,48 as well as spatially in the heart wall46. With these values, we can estimate that c = 5.19 × 10−5 m2 and τ = 0.23 s. This result suggests that, while the shrinking of the in-plane stochastic bias is exponential, it seems plausible that at least some effect on the wave front due to the anisotropy can persist for a non-trivial fraction of the heartbeat. Uncertainty in the measurement of diffusivities as well as their spatial variation preclude surety in our conclusion. After approximately τ = 0.23 seconds have passed, the ratio of the magnitude of the x (or y) component of the first term in the variance equation to the magnitude of the x (or y) component of its second term, is 2.63, meaning that the first (isotropic) term dominates.

In summary, based on our results concerning the drift vector, we restricted our attention to the GHM with vanishing in-plane curvatures, and considered the variance of the diffusion process, which can serve as a measure of its growth rate along each dimension. We showed the stochastic bias of this process in the heart wall tangent plane vanished exponentially quickly, as long as k B  ≠ 0. In combination with our results showing the vanishing of the deterministic bias (i.e. Ito drift), the shrinking of the anisotropy of the variance vector in the heart wall tangent plane suggests that the helicoidal fiber geometry found in mammalian hearts4,33, where k N , k T  ≈ 0 and k B  ≠ 0, minimizes the diffusion bias.

## Discussion

The contraction of myocytes in the heart wall is controlled by current flowing through an elaborate Purkinje network, which raises the question of how current diffuses through the wall around the insertion points. Classically, the propagation of the contraction signal is represented using a reaction-diffusion equation with the inverse diffusivity tensor as its metric17,27,28. We combined this Riemannian approach with a minimal surface model of the cardiac fiber geometry, the generalized helicoid model (GHM)4, and exploited the Ito calculus to describe and analyze diffusion on it.

Among our results, we showed that the helicoidal structure of cardiac fibers minimizes local biases in the diffusion, both deterministic (embodied by the Ito drift vector) and stochastic (derived from differing x and y components of the second moment of the Ito diffusion SDEs). Since the GHM is a local approximation anywhere in the left ventricular wall4, one would expect it to not favor biased flow in a particular direction. In line with these theoretical expectations, we find that the GHM with k N  = k T  = 0 minimizes the Ito drift, providing a natural interpretation for the recent imaging studies that suggest |k N | and |k T | are relatively small across species4,33. Moreover, using the variance of the Ito diffusion on the cardiac manifold with vanishing in-plane curvatures, we also show that stochastic bias in the heart wall tangent plane shrinks exponentially fast in time, becoming isotropic with a rate given by averaging the squares of the speeds along and across fibers.

This solution is also good in a functional analytic sense, for both the drift and Ricci curvature. Using the calculus of variations, we showed that the GHM lies at a stationary point of the Ricci curvature in the space of orientation functions, provided the in-plane curvatures are zero. However, it is neither a minimum nor a maximum. One might speculate that this is because increased negativity of the Ricci scalar, equivalent to increased transmural rotation rate (or |k B | for the GHM) could lead to cardiomyopathic disturbances in the signal wavefront as it propagates26, such as scroll wave turbulence22,23. On the other hand, there are a number of potential mechanical reasons for the presence of the helicoidal structure, including shear wave filtering49,50, energy dissipation6, and mechanical efficiency11,29. Thus, the competing effects of changing k B must be balanced to allow safe and fast wave propagation, as well as mechanical efficacy. Physical constraints, such as the myocyte size and necessary accompanying extracellular matrix structure, which idealized models often do not capture, also affect the structure.

While we have considered a number of properties conferred upon the heart by its helicoidal geometry, there are a few limitations inherent to our approach. Currently, we consider only local, static fiber geometry within the left ventricle. Our analysis also only considers the diffusion term in the reaction-diffusion contraction wave propagation equation, since this is the term that is affected by myocyte orientation and it largely governs the local spreading of the signal. In addition, we do not consider the effect of cardiomyocyte laminar sheets, which may contribute to both the electrophysiology17,51 and mechanics52 of the heart. Our approach provides useful insights, despite these approximations, because the effect of the fibers on diffusion of the contraction signal has been empirically shown to be much greater than that of the sheets35. Lastly, the GHM assumes that fibers lie in planes tangent to the heart wall. Although this is an acceptable local approximation4,33, some studies suggest that the presence of fibers with an out-of-plane component may play a role in cardiac function53. Finally, exploring the link between local diffusion bias and long-term wave propagation in the heart is a promising avenue for future work.

In the end, our analysis reveals that the GHM arrangement minimizes any in-plane directional biases in the diffusion process, provided that there is transmural rotation and small in-plane curvature of the fibers in the heart wall. Normally, one might have thought about the orientation of fibers from a purely mechanical perspective. However, given the ubiquity of helicoidal arrangements in biology and artificial composites5,6,7,8,9, it should perhaps not be a surprise that nature has used this geometry to satisfy both the mechanical and electrical requirements of the heart.

## References

1. 1.

Ten Tusscher, K. & Panfilov, A. Modelling of the ventricular conduction system. Prog. Biophys. Mol. Biol. 96, 152–170 (2008).

2. 2.

Durrer, D. et al. Total excitation of the isolated human heart. Circ. 41, 899–912 (1970).

3. 3.

Anderson, R. H., Smerup, M., Sanchez-Quintana, D., Loukas, M. & Lunkenheimer, P. P. The three-dimensional arrangement of the myocytes in the ventricular walls. Clin. Anat. 22, 64–76 (2009).

4. 4.

Savadjiev, P. et al. Heart wall myofibers are arranged in minimal surfaces to optimize organ function. Proc. Natl. Acad. Sci. 109, 9248–9253 (2012).

5. 5.

Neville, A. C. Biology of fibrous composites: development beyond the cell membrane (Cambridge University Press, 1993).

6. 6.

Weaver, J. C. et al. The stomatopod dactyl club: a formidable damage-tolerant biological hammer. Sci. 336, 1275–1280 (2012).

7. 7.

Zimmermann, E. A. et al. Mechanical adaptability of the Bouligand-type structure in natural dermal armour. Nat. communications 4 (2013).

8. 8.

Apichattrabrut, T. & Ravi-Chandar, K. Helicoidal composites. Mech. Adv. Mater. Struct. 13, 61–76 (2006).

9. 9.

Grunenfelder, L. et al. Bio-inspired impact-resistant composites. Acta biomaterialia 10, 3997–4008 (2014).

10. 10.

LeGrice, I., Pope, A. & Smaill, B. The architecture of the heart: Myocyte organization and the cardiac extracellular matrix. In Interstitial Fibrosis in Heart Failure, 3–21 (Springer, 2005).

11. 11.

Vendelin, M., Bovendeerd, P. H., Engelbrecht, J. & Arts, T. Optimizing ventricular fibers: uniform strain or stress, but not atp consumption, leads to high efficiency. Am. J. Physiol. Circ. Physiol. 283, H1072–H1081 (2002).

12. 12.

Ingels, N. B. Jr Myocardial fiber architecture and left ventricular function. Technol. Heal. Care 5, 45–52 (1997).

13. 13.

Bettex, D. A., Prêtre, R. & Chassot, P.-G. Is our heart a well-designed pump? the heart along animal evolution. Eur. heart journal 35, 2322–2332 (2014).

14. 14.

Franzone, P. C., Guerri, L., Pennacchio, M. & Taccardi, B. Spread of excitation in 3-d models of the anisotropic cardiac tissue. ii. effects of fiber architecture and ventricular geometry. Math. biosciences 147, 131–171 (1998).

15. 15.

Bernus, O., Wellner, M. & Pertsov, A. Intramural wave propagation in cardiac tissue: asymptotic solutions and cusp waves. Phys. Rev. E 70, 061913 (2004).

16. 16.

Clerc, L. Directional differences of impulse spread in trabecular muscle from mammalian heart. The J. physiology 255, 335–346 (1976).

17. 17.

Young, R. J. & Panfilov, A. V. Anisotropy of wave propagation in the heart can be modeled by a Riemannian electrophysiological metric. Proc. Natl. Acad. Sci. 107, 15063–15068 (2010).

18. 18.

Castro-Villarreal, P. Brownian motion meets Riemann curvature. J. Stat. Mech. Theory Exp. 2010, P08006 (2010).

19. 19.

Liao, M. & Zheng, W. Radial part of Brownian motion on a Riemannian manifold. The Annals Probab. 173–177 (1995).

20. 20.

Yoshigaki, T. Theoretically predicted effects of Gaussian curvature on lateral diffusion of membrane molecules. Phys. Review E 75, 041901 (2007).

21. 21.

Faraudo, J. Diffusion equation on curved surfaces. i. theory and application to biological membranes. The J. chemical physics 116, 5831–5841 (2002).

22. 22.

Fenton, F. & Karma, A. Fiber-rotation-induced vortex turbulence in thick myocardium. Phys. review letters 81, 481 (1998).

23. 23.

Fenton, F. & Karma, A. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos: An Interdiscip. J. Nonlinear Sci. 8, 20–47 (1998).

24. 24.

Xie, F. et al. A simulation study of the effects of cardiac anatomy in ventricular fibrillation. The J. clinical investigation 113, 686–693 (2004).

25. 25.

Qu, Z., Kil, J., Xie, F., Garfinkel, A. & Weiss, J. N. Scroll wave dynamics in a three-dimensional cardiac tissue model: roles of restitution, thickness, and fiber rotation. Biophys. J. 78, 2761–2775 (2000).

26. 26.

Ripplinger, C. M. et al. Enhanced transmural fiber rotation and connexin 43 heterogeneity are associated with an increased upper limit of vulnerability in a transgenic rabbit model of human hypertrophic cardiomyopathy. Circ. research 101, 1049–1057 (2007).

27. 27.

Wellner, M., Berenfeld, O., Jalife, J. & Pertsov, A. M. Minimal principle for rotor filaments. Proc. Natl. Acad. Sci. 99, 8015–8018 (2002).

28. 28.

Ten Tusscher, K. & Panfilov, A. Eikonal formulation of the minimal principle for scroll wave filaments. Phys. review letters 93, 108106 (2004).

29. 29.

Rijcken, J., Bovendeerd, P., Schoofs, A., Van Campen, D. & Arts, T. Optimization of cardiac fiber orientation for homogeneous fiber strain during ejection. Annals biomedical engineering 27, 289–297 (1999).

30. 30.

Thomas, C. E. The muscular architecture of the ventricles of hog and dog hearts. Am. J. Anat. 101, 17–57 (1957).

31. 31.

Streeter, D. D., Spotnitz, H. M., Patel, D. P., Ross, J. & Sonnenblick, E. H. Fiber orientation in the canine left ventricle during diastole and systole. Circ. research 24, 339–347 (1969).

32. 32.

Bayer, J., Blake, R., Plank, G. & Trayanova, N. A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models. Annals biomedical engineering 40, 2243–2254 (2012).

33. 33.

Piuze, E., Sporring, J. & Siddiqi, K. Maurer-Cartan forms for fields on surfaces: application to heart fiber geometry. Pattern Analysis Mach. Intell. IEEE Transactions on 37, 2492–2504 (2015).

34. 34.

Pravdin, S. F. et al. Electrical wave propagation in an anisotropic model of the left ventricle based on analytical description of cardiac architecture. PLoS One 9, e93617, https://doi.org/10.1371/journal.pone.0093617 (2014).

35. 35.

Caldwell, B. J. et al. Three distinct directions of intramural activation reveal nonuniform side-to-side electrical coupling of ventricular myocytes. Circ. Arrhythmia Electrophysiol. 2, 433–440 (2009).

36. 36.

Hsu, P. Brownian motion and Riemannian geometry. Contemp. Math 73, 95–104 (1988).

37. 37.

Itô, K. The Brownian motion and tensor fields on Riemannian manifold. Proc. Int. Congr. Math., Stock. (1962).

38. 38.

Hsu, E. P. Stochastic analysis on manifolds, vol. 38 (American Mathematical Soc., 2002).

39. 39.

Fuster, A., Tristan-Vega, A., Haije, T. D., Westin, C.-F. & Florack, L. A novel Riemannian metric for geodesic tractography in dti. In Computational Diffusion MRI and Brain Connectivity, 97–104 (Springer, 2014).

40. 40.

Fuster, A. et al. Adjugate diffusion tensors for geodesic tractography in white matter. J. Math. Imaging Vis. 54, 1–14 (2016).

41. 41.

Hsu, E. P. Heat equations on manifolds and Bismut’s formula. Contemp. Math. 429, 121 (2007).

42. 42.

Ikeda, N. & Watanabe, S. Stochastic differential equations and diffusion processes, vol. 24 (Elsevier, 2014).

43. 43.

Gray, A. et al. The volume of a small geodesic ball of a Riemannian manifold. The Mich. Math. J. 20, 329–344 (1974).

44. 44.

Uemura, H. et al. On a short time expansions of the fundamental solution of heat equations by the method of wiener functional. J. Math. Kyoto Univ. 27, 417–431 (1987).

45. 45.

Gelfand, I. & Fomin, S. Calculus of Variations (Courier Corporation, 2012).

46. 46.

Kléber, A. G. & Rudy, Y. Basic mechanisms of cardiac impulse propagation and associated arrhythmias. Physiol. reviews 84, 431–488 (2004).

47. 47.

Costa, C. M., Hoetzl, E., Rocha, B. M., Prassl, A. J. & Plank, G. Automatic parameterization strategy for cardiac electrophysiology simulations. In Computing in Cardiology Conference (CinC), 2013, 373–376 (IEEE, 2013).

48. 48.

Neic, A. et al. Efficient computation of electrograms and ecgs in human whole heart simulations using a reaction-eikonal model. J. computational physics 346, 191–211 (2017).

49. 49.

Guarn-Zapata, N., Gomez, J., Yaraghi, N., Kisailus, D. & Zavattieri, P. D. Shear wave filtering in naturally-occurring Bouligand structures. Acta biomaterialia 23, 11–20 (2015).

50. 50.

Lakhtakia, A. Wave propagation in a piezoelectric, continuously twisted, structurally chiral medium along the axis of spirality. Appl. Acoust. 44, 25–37 (1995).

51. 51.

Hooks, D. A. et al. Laminar arrangement of ventricular myocytes influences electrical behavior of the heart. Circ. Res. 101, e103–e112 (2007).

52. 52.

Holzapfel, G. A. & Ogden, R. W. Constitutive modelling of passive myocardium: a structurally based framework for material characterization. Philos. Transactions Royal Soc. Lond. A: Math. Phys. Eng. Sci. 367, 3445–3475 (2009).

53. 53.

Lunkenheimer, P. P. et al. Three-dimensional architecture of the left ventricular myocardium. The Anat. Rec. Part A: Discov. Mol. Cell. Evol. Biol. 288, 565–578 (2006).

54. 54.

Rößler, A. Runge–kutta methods for the strong approximation of solutions of stochastic differential equations. SIAM J. on Numer. Analysis 48, 922–952 (2010).

55. 55.

Matthew, A. Critical fluctuations and coupling of stochastic neural mass models. Ph.D. thesis, The University of Queensland. Critical fluctuations and coupling of stochastic neural mass models (2017).

## Acknowledgements

We thank Rustum Choksi for helpful discussions and for guidance on the variational calculations.

## Author information

Authors

### Contributions

T.A., P.S. and K.S. conceived idea; T.A., A.K., S.W.Z. and K.S. designed and carried out research; T.A., S.W.Z. and K.S. wrote the paper.

### Corresponding author

Correspondence to Kaleem Siddiqi.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Aumentado-Armstrong, T., Kadivar, A., Savadjiev, P. et al. Conduction in the Heart Wall: Helicoidal Fibers Minimize Diffusion Bias. Sci Rep 8, 7165 (2018). https://doi.org/10.1038/s41598-018-25334-7

• Accepted:

• Published:

• ### Ex vivo cardiovascular magnetic resonance diffusion weighted imaging in congenital heart disease, an insight into the microstructures of tetralogy of Fallot, biventricular and univentricular systemic right ventricle

• Cyril Tous
• , Thomas L. Gentles
• , Alistair A. Young
•  & Beau P. Pontré

Journal of Cardiovascular Magnetic Resonance (2020)

• ### (INVITED) Reaction–diffusion waves in cardiovascular diseases

• A.V. Panfilov
• , H. Dierckx
•  & V. Volpert

Physica D: Nonlinear Phenomena (2019)

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


text:
Discriminating

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

Factorisable Quadratics

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.

Which Quadratic?

In this activity you will need to work in a group to connect different representations of quadratics.

Powerful Quadratics

Age 16 to 18
Thank you to Maria from Hills Road Sixth Form College, Eleanor from The King's School in Macclesfield, Abi, and Rhombusta for submitting solutions to this problem!

Here's Abi's solution:

For equations of the form $(ax^2+bx+c)^{(dx^2+ex+f)}=1$, where $a,b,c,d,e,f$ and $x$ are real, there are three possibilities:
• $ax^2+bx+c=1$, as $1^n=1$ for any $n$
• $dx^2+ex+f=0$, as $n^0=1$ for any $n$
• $ax^2+bx+c=-1$, and $dx^2+ex+f$ is an even integer, as $(-1)^{(2n)}=1$ for any integer $n$.
Now to solve the problems:

$i)$

Case $1$: $x^2-7x+11=1$ factorises as $(x-5)(x-2)=0$ so $x=5$ or $x=2$.
Case $2$: $x^2-11x+30=0$ factorises as $(x-5)(x-6)=0$, so $x=6$ or $x=5$.
Case $3$: $x^2-7x+11=-1$ factorises as $(x-3)(x-4)=0$. As $3^2-11(3)+30=6$ and $4^2-11(4)+30=2$ are both even, $x=3$ or $x=4$ are solutions.

So the solutions are $x=2,3,4,5$ and $6$.

$ii)$

Case $1$: $2-x^2=1$ factorises as $(1-x)(1+x)=0$ so $x=1$ or $x=-1$.
Case $2$: $x^2-3 \sqrt{2}x+4=0$ factorises as $(x-2 \sqrt{2})(x- \sqrt{2})$, so $x=2 \sqrt{2}$ or $x = \sqrt{2}$.
Case $3$: $2-x^2=-1$ factorises as $(\sqrt{3}-x)(\sqrt{3}-x)$, but as $\sqrt{3}^2-3 \sqrt{2} \sqrt{3} +4$ and $\sqrt{3}^2-3 \sqrt{2} \sqrt{3} + 4$ are not even integers, these are invalid.

So the solutions are $x=-1,1, \sqrt{2}$ and $2 \sqrt{2}$.

format: plain


text:
# Numpy Multivariate Normal Pdf

patches import Ellipse def plot_point_cov(points, nstd=2, ax=None, **kwargs): """ Plots an nstd sigma ellipse based on the mean and covariance of a point "cloud" (points, an Nx2 array). I've not had time or need yet to fix it up. the log likelihood of a set of datapoints with respect to the probability distribution. Isromi Janwar. Cumulative Distribution Function of the Standard Normal Distribution. Bayesian inference for the multivariate Normal is most simply instanti-ated using a Normal-Wishart prior over the mean and covariance. In Python it is possible to generate values following a truncated normal distribution with the scipy. построение 3d-рассеяния в matplotlib. Moving between nodes of the tree, ac-complished by the FCMdata. Creates a multivariate normal (also called Gaussian) distribution parameterized by a mean vector and a covariance matrix. edu Summary Objective Bayesian inference for the multivariate normal distribution is il-. logcdf rv_continuous. This chapter of the tutorial will give a brief introduction to some of the tools in seaborn for examining univariate and bivariate distributions. multivariate_normal. Multivariate Data Analysis (7th Edition) KEY BENEFIT: For over 30 years, this text has provided students with the information they need to understand and apply multivariate data analysis. 1 Testing the multivariate Gaussian PDF implementation. samples_generator import make_blobs import numpy as np from scipy. 0, size=None) ¶ Draw samples from a log-normal distribution. Kernel Density Estimation is a method to estimate the frequency of a given value given a random sample. take a sample from uniform distribution and put the value in the function found. The probability of landing in a narrow range of width d between x d/2 and x+d/2 is about p(x)d. Visualizing the distribution of a dataset¶ When dealing with a set of data, often the first thing you'll want to do is get a sense for how the variables are distributed. It represents the distribution of a multivariate random variable that is made up of multiple random variables that can be correlated with eachother. Time Series. pdf that computes the likelihood of seeing a data point in a multivariate Gaussian distribution. Replaces missing values with random draws from this distribution. random import standard_normal, chisquare, multivariate_normal, dirichlet, multinomial from numpy. numpy関数numpy. Normal random numbers can also be generated using the general inverse transform method (e. Welcome to Statsmodels’s Documentation¶. Area shows probability between (a1~a2) and (b1~b2) in bivariate normal distribution - contour. The search for the optimal penalization parameter (alpha) is done on an iteratively refined grid: first the cross-validated scores on a grid are computed, then a new refined grid is centered around the maximum, and so on. And covariance parameters, returning a frozen multivariate normal. For more on simulating a draw from the truncated normal distribution, see Robert (1995), Lynch (2007) Section 8. edu James O. We will not be using NumPy in this post, but will do later. And then we calculate f1score to find out a value (Epsilon) which we can set as best decisive threshold between our normal and abnormal values. This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Regression 2. The following are code examples for showing how to use scipy. multivariate_normal¶ cupy. JoelCornett i dont think that is the multivariate normal PDF pyCthon. 8 Further Reading 6. Is there any python package that allows the efficient computation of the multivariate normal pdf? It doesn't seem to be included in Numpy/Scipy, and surprisingly a Google search didn't turn up any useful thing. Most of the models Bishop develops use that as a foundational framework, and a big part of that has to do with treating the covariance matrix as a block matrix and figuring out conditional/marginal distributions given a joint. com, Charlottesville, United States ABSTRACT OncoLnc is a tool for interactively exploring survival correlations, and for. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. In general it is best to use existing implementations of stuff like this - this post is just a learning exercise. Why the Simple Regression Model is Not Enough By now we know how to explore the relationship between a dependent and an independent variable through regression analysis. stax) bernoulli() (in module jax. rand(5) mvnorm. # Assume a mean of 0 for simplicity ys = np. Title changed from need errors for non-physical numpy. The likelihood of a point being generated by a cluster will be evaluated using a multivariate Gaussian distribution, as was done in class. Kernel density estimation using Python, matplotlib. Another way to generate random numbers or draw samples from multiple probability distributions in Python is to use NumPy’s random module. Eine multivariate Verteilung ist in der Wahrscheinlichkeitsrechnung und in der Statistik die Verteilung eines Zufallsvektors, also einer Zufallsvariablen, deren Werte Vektoren im sind. Statistical. 3 Multivariate Analysis 48 2. y = mvnpdf(X) returns an n-by-1 vector y containing the probability density function (pdf) of the d-dimensional multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of the n-by-d matrix X. Here we list some scipy related pdf books, and you can choose the most suitable one for your needs. By voting up you can indicate which examples are most useful and appropriate. I know it is a joke to do get it from there, but nevertheless. multivariate_normal(mean, cov [, size])¶ Draw random samples from a multivariate normal distribution. It looks normal, but it\'s skewed. Returns a single numpy. multivariate_normal; it is then fed to the hist2d function of pyplot matplotlib. multivariate_normal¶ scipy. I am providing this information to give you a general idea of the pace and timing of the class. I shall present here code and examples for doing this with SAS and with R. stats import multivariate_normalimport numpy as np … - Selection from SciPy Recipes [Book]. Properties of the multivariate Gaussian probability distribution. Probability density function. - animation. This question is rather basic, but I can\'t seem to find the answer for R on the internet. Compatibility Layers¶. Ossama Embarak. Rather than creating a single subplot, this function creates a full grid of subplots in a single line, returning them in a NumPy array. SAS The code below will simulate data for a matrix of correlations between variables Y1, Y2, Y3,. Exercise 6 (The bivariate normal distribution). samples_generator import make_blobs import numpy as np from scipy. See also cupy. However, these terms actually represent 2 very distinct types of analyses. This website uses cookies to ensure you get the best experience on our website. To describe them, a normal distribution with mean in both x and y is needed. function multivariate normal in numpy. Normalization in this sense means to transform a vector so that it has a length of one. The Exponential distribution is the continuous analog of the Geometric distribution. The py-earth package implements Multivariate Adaptive Regression Splines using Cython and provides an interface that is compatible with scikit-learn's Estimator, Predictor, Transformer, and Model interfaces. We use cookies for various purposes including analytics. Now we can take this covariance matrix and pipe it into the numpy PDF routine to get our distribution: from scipy. 8] group_sigmas = [1. Mathematical Details. This post is mainly some notes about linear algebra, the cholesky decomposition, and a way of parametrising the multivariate normal which might be more efficient in some cases. They are extracted from open source Python projects. Thanks for watching!!  //Z table link https://drive. Available online 9 November 2019, 112971. So let's first talk about a probability density function. The normal may be univariate or multivariate. The truncnorm package in R also has functions to draw from a truncated normal. al provides an applications-oriented introduction to multivariate analysis for the non-statistician. This example doesn’t use any input signals u. Multivariate Normal Distribution In this lesson we discuss the multivariate normal distribution. One of these is to implement the probability density function (PDF) for a multivariate normal (MVN) distribution. JoelCornett i dont think that is the multivariate normal PDF pyCthon. To compute this part, the scipy package provides a convenient function multivariate_normal. y = mvnpdf(X) returns an n-by-1 vector y containing the probability density function (pdf) of the d-dimensional multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of the n-by-d matrix X. pdf(x) #evaluate the probability density at x. It's something that works. Learn more about cumsum, for loop, cumulative, sum. 前回は、1次元ガウス分布のベイズ学習について勉強しました。 今回は、多次元ガウス分布について勉強します。 （今回も教材として「ベイズ推論による機械学習」を使用します。） 多次元ガウス分布 1次元ガウス分布. Statistics with SciPy import numpy as np In [30]: np. optimize sub-package. multivariate Dark theme Light theme #lines '''Multivariate Distribution Probability of a multivariate t distribution Now also mvstnormcdf has tests against R mvtnorm Still need non-central t, extra options, and convenience function for location, scale version. Note that in the multiparameter case I(θ) is a matrix so“inverse Fisher information”involves a matrix inverse. get_window, etc. lognormal¶ numpy. The most well-known tool to do this is the histogram. Is this a valid algorithm for generating a 3D Wiener process? (When I graph the results, they certainly look like potential Brownian. optimize import minimize. scipy gaussian Alternatively, freeze the distribution and display the frozen pdf: numpy. We motivate the use of symbolics and thin compilers in scientiﬁc computing. Normal data is generated by a multivariate Gaussian distribution and outliers are generated by a uniform distribution. multivariate_normal taken from open source projects. y = normpdf(x) returns the probability density function (pdf) of the standard normal distribution, evaluated at the values in x. (i)Use numpy. The article is a development of our research on estimation of multivariate probit models (Cappellari and Jenkins, 2003). This is an excerpt from the Python Data Science Handbook by Jake VanderPlas; Jupyter notebooks are available on GitHub. In statistics, a mixture model is a probabilistic model for density estimation using a mixture distribution. Miller⁄ Mathematics Department Brown University Providence, RI 02912 Abstract The Method of Least Squares is a procedure to determine the best ﬁt line to data; the proof uses simple calculus and linear algebra. If you are interested in an instructor-led classroom training course, you may have a look at the Python classes. stats implementation of a multivariate normal distribution: from scipy. multiply and other ufuncs. Finding the probabilities from multivariate normal distributions. They are extracted from open source Python projects. Time Series. Returns m numpy. The multivariate normal, multinormal or Gaussian distribution is a numpy. One of the many sub-problems to tackle is writing a function that calculates the probability density function (PDF) for a multivariate normal (MVN) distribution. The py-earth package implements Multivariate Adaptive Regression Splines using Cython and provides an interface that is compatible with scikit-learn's Estimator, Predictor, Transformer, and Model interfaces. Monte Carlo methods for risk analysis. pdf(x) #evaluate the probability density at x. The Central Idea Behind Generative Classification. A simple PDF and the PDF for a normal probability distribution with mean = 0. If Y is a random variable with a normal distribution, and P is the standard logistic function, then X = P(Y) has a logit-normal distribution; likewise, if X is logit-normally distributed, then Y = logit(X)= log (X/(1-X)) is normally distributed. In statistics, a mixture model is a probabilistic model for density estimation using a mixture distribution. models import Normal from tensorflow. 0 and sd = 1. Vectorization Python Numpy is a well-known and reliable vectorized linear algebra library which is a foundation of scientific ( SciPy ) and machine learning. multivariate_normal. • The general multivariate normal density the probability density function, but is interpreted as a Maximum A-Posteriori (MAP) Estimation \$ () 0 0 1. Krishnamoorthy, K. logpdf which don’t support support the allow_singular keyword prior to verion 0. Values along the diagonal are independent variance terms for each dimension; values off the diagonal are interaction terms (lead to 'rotated' Gaussians, frex). I observed an odd behavior of the multivariate_normal function, which does not. In this tutorial, we're going to show a Python-version of kernels, soft-margin, and solving the quadratic programming problem with CVXOPT. NumPy Intermediate (Numerical Python) arrays, boolean arrays, masks, broadcasting, fancy indexes, sorting, structured data, aggregations, ufuncs, datatypes. However, in the case of a probability density function, these numbers provide little or no insight into the distribution of samples. multivariate_normal(mean, cov [, size])¶ Draw random samples from a multivariate normal distribution. \end{align} Related distributions. First use Numpy. For example, training a Scipy's gaussian_kde with a numpy array of shape (10000, 2) is interpreted as two instances of 10000 dimensions. 6 Density of the Multivariate Normal 6. You can vote up the examples you like or vote down the ones you don't like. The cdf function of multivariate_normal may not have the correct results when the dimension is 2. pdf函数,但不断出错. sf rv_continuous. A simple histogram can be a great first step in understanding a dataset. Even explaining what that means is quite a challenge. Krishnamoorthy, K. The Multivariate Gaussian Distribution Chuong B. distributions. They are extracted from open source Python projects. The parent object for a matrix is a matrix space MatrixSpace(R, n, m)of all × matrices over a ring. Alternately, the distribution may be exponential, but may look normal if the observations are transformed by taking the natural logarithm of the values. The results were a bit surprising because MATLAB hardly produced a 1. Generator in numpy 1. bivariate_normalの存在を知って使っていたのだけれど，どうもおかしいということに気づいて，引数の意味を勘違いしていたのでメモ．API Documentを眺めるとこんな感じ…. random import normal from scipy. numpy関数numpy. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Parameters x array_like. For that, we can use MultivariateNormalParameter, which creates a parameter that has a multivariate normal posterior, with full covariance. The Multivariate Normal distribution is defined over R^k and parameterized by a (batch of) length-k loc vector (aka 'mu') and a (batch of) k x k scale matrix; covariance = scale @ scale. † Furthermore, because X and Y are linear functions of the same two independent normal random variables, their joint PDF takes a special form, known as the bi-variate normal PDF. The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. pdf(r) # calculate the PDF of all values r at the same time # Log of Gaussian PDF is much faster to calculate, since there are no exponentials # adding is a cheaper. (Bot-tom row) A histogram plot for the posterior distribution of ˆbased upon the samples in the chain. Miller⁄ Mathematics Department Brown University Providence, RI 02912 Abstract The Method of Least Squares is a procedure to determine the best ﬁt line to data; the proof uses simple calculus and linear algebra. multivariate_normal. 17+ and automatically fall back to numpy. Changing the step size (e. multivariate_normalがあり、これはMATLABのmvnpdfと同じことができると思われますが、私はdocumentationを理解できないように苦労しています。. Is there a reason why pm. It looks normal, but it\'s skewed. metrics import f1_score As shown in the figure, we can see the datapoints tightly clustered at the centre with some few points further away from the cluster. stats import norm from scipy. stats import multivariate_normal from random import shuffle from tqdm import tqdm_notebook as tqdm. For installing these amazing packages there are tons of resources on the web, we just point at Installing the SciPy Stack. This comes up e. T where @ denotes matrix-multiplication. 尤其是在获得numpy. normgen object at. By voting up you can indicate which examples are most useful and appropriate. pdf (input_values, x) [source] Calculates the probability density function of the model. pdf (sample, mean, cov) ¶ Return a theano expression representing the values of the probability density function of the multivariate normal. stats import multivariate_normal. The multivariate normal distribution in general. Please consider adding the complex multivariate normal distribution. A simple PDF and the PDF for a normal probability distribution with mean = 0. Vectorization Python Numpy is a well-known and reliable vectorized linear algebra library which is a foundation of scientific ( SciPy ) and machine learning. stats import multivariate_normal P_0 = np. Module containing expression buildes for the multivariate normal. Statistical. The article is a development of our research on estimation of multivariate probit models (Cappellari and Jenkins, 2003). Quantiles, with the last axis of x denoting the components. isf rv_continuous. Continuous. Second-order complex random vectors and normal distributions. So let's first talk about a probability density function. One thing that may be troubling is the requirement for a prior. y = mvnpdf(X) returns an n-by-1 vector y containing the probability density function (pdf) of the d-dimensional multivariate normal distribution with zero mean and identity covariance matrix, evaluated at each row of the n-by-d matrix X. 1 Standard Normal: The Basics 18. By voting up you can indicate which examples are most useful and appropriate. Since our model will usually contain a constant term, one of the columns in the X matrix will contain only ones. stats import multivariate_normal as mvnorm x = np. multivariate normal. The marginal distribution of a multivariate normal random variable is itself normal. Thanks for watching!!  //Z table link https://drive. While in the previous section we restricted our attention to the multivariate normal distribution with zero mean and unit covariance, we now deal with the general case. stats import multivariate_normal. Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. To make this concrete, below is an example of a sample of Gaussian numbers transformed to have an exponential distribution. mvtol_region (x, alpha, P, B) [source] ¶ Computes tolerance factor for multivariate normal. If you wanted to know how three variables. The default is window_hanning. multivariate normal probability density function. functions yourself. Struggling to create the contour plot of a 2d Normal Distribution. pdf (…) einen Wert außerhalb von [0,1] unerwartet produzieren” Ein PDF kann grundsätzlich jeden nicht-negativen Wert einnehmen, solange er sich in einen integriert. ECE595 Homework 1 Linear Algebra and Probability Review 1 Guanzhe Hong Spring 2019 School of ECE, Purdue University. JoelCornett i dont think that is the multivariate normal PDF pyCthon Jul 23 12 at 15: 41. Computing the probability of success for our new Employee is actually pretty simple using numpy in Python. set from scipy. Readers with previous exposure to likelihood theory may have a few ques-tions here, mostly about n. import numpy as np def ''' # generate proposal state ProposalParameters = np. Distribution of Xi is multivariate normal with mean vector Ui and variance covariance matrix sigma Ii. Multivariate regression technique can be implemented efficiently with the help of matrix operations. Most of the models Bishop develops use that as a foundational framework, and a big part of that has to do with treating the covariance matrix as a block matrix and figuring out conditional/marginal distributions given a joint. 1BestCsharp blog 3,713,963 views. 5 Diagonals in Hyperspace 6. multivariate_normalがあり、これはMATLABのmvnpdfと同じことができると思われますが、私はdocumentationを理解できないように苦労しています。. normal(): 正規分布（任意の平均、標準偏差） np. 14, there has been a multivariate_normal function in the scipy. A probability density function (pdf) is a function that can predict or show the mathematical probability of a value occurring between a certain interval in the function. Bayesian inference for the multivariate Normal is most simply instanti-ated using a Normal-Wishart prior over the mean and covariance. Numpy has a random. Mean of the distribution (default zero) Covariance matrix of the distribution (default one) Whether to allow a singular covariance matrix. For a standard PDF #!python import numpy as np from fastkde import fastKDE import pylab as PP #Generate two random variables dataset (representing 100000 pairs of datapoints) N = 2e5 var1 = 50*np. pdf that computes the likelihood of seeing a data point in a multivariate Gaussian distribution. pomegranate has been shown to be faster than numpy at updating univariate and multivariate gaussians. - Sparse matrices are no longer limited to 2^31 nonzero elements. Learn More. This website uses cookies to ensure you get the best experience on our website. This task is called density estimation. Finding the probabilities from multivariate normal distributions. multivariate_normal function from numpy. edu is a platform for academics to share research papers. The joint probability density function (joint pdf) is a function used to characterize the probability distribution of a continuous random vector. We have discussed a single normal random variable previously; we will now talk about two or more normal random variables. 此主题已被删除。只有拥有主题管理权限的用户可以查看。. multivariate_normal. Alternatively, the object may be called (as a function) to fix the mean. stats import multivariate_normal import warnings warnings. To compute this part, the scipy package provides a convenient function multivariate_normal. y = normpdf( x , mu ) returns the pdf of the normal distribution with mean mu and the unit standard deviation, evaluated at the values in x. Probability distribution value. For more on simulating a draw from the truncated normal distribution, see Robert (1995), Lynch (2007) Section 8. MULTIVARIATE NORMAL DISTRIBUTION pdf book, 2. Python標準にも random というモジュールがあるが、ベクトル演算の可能な numpy のほうが「大量に乱数を生成してなんかの処理をする」という場合に高速に動く。あと分布関数が山ほど用意さ. The top equation is the PDF for a Normal distribution with a single X variable. Given a set of observations $$(x_i)_{1\leq i \leq n}$$. ECE595 Homework 1 Linear Algebra and Probability Review 1 Guanzhe Hong Spring 2019 School of ECE, Purdue University. lognormal (mean=0. Data with this distribution is called log-normal. Calculation of the probability density function in python. The text is released under the CC-BY-NC-ND license, and code is released under the MIT license. Though it is possible to now draw new vector examples easily using the numpy function numpy. pyplot as plt # Define numbers of generated data points and bins per axis. 1 Random vectors Deﬁnition 3. OLS in Matrix Form 1 The True Model † Let X be an n £ k matrix where we have observations on k independent variables for n observations. Scipy gaussian pdf A normal continuous random variable. Analysis_of_Multivariate_and_High-Dimen. numpy - How to weigh a function with 2 variables with a Gaussian distribution in python? statistics - Fitting Gaussian KDE in numpy/scipy in Python; numpy - Python: fit data with gaussian rising and exponential decay; python - Drawing from certain probabilities in Gaussian Normal Multivariate Distribution in numpy. dev3478 1 By using OBJECT arrays, one can effectively have heterogeneous arrays, but the system still sees each element of the array as exactly the same thing (a reference. For example, if I would like to draw 2 samples, I tried. The truncnorm package in R also has functions to draw from a truncated normal. stats import multivariate_normal P_0 = np. 1 Testing the multivariate Gaussian PDF implementation. The Multivariate Gaussian Distribution Chuong B. (experimental) Returns an array of samples drawn from the multivariate normal distribution. multivariate_normal. The PDF is well-defined for multivariate distributions. random import multivariate_normal data = np. Time Series. NumPy / SciPy Recipes for Data Science: k-Medoids Clustering Christian Bauckhage B-IT, University of Bonn, Germany Fraunhofer IAIS, Sankt Augustin, Germany Abstract—In this note, we study k. The mean keyword specifies the mean. Monte Carlo methods for risk analysis. In the context of a Gibbs sampler, I profiled my code and my major bottleneck is the following:. Learning Deep Broadband [email protected] Hongjoo LEE 2. stats import multivariate_normalimport numpy as np … - Selection from SciPy Recipes [Book]. stats import multivariate_normal as mulnorm import matplotlib. The multivariate normal distribution in general. For example, training a Scipy's gaussian_kde with a numpy array of shape (10000, 2) is interpreted as two instances of 10000 dimensions. The py-earth package implements Multivariate Adaptive Regression Splines using Cython and provides an interface that is compatible with scikit-learn's Estimator, Predictor, Transformer, and Model interfaces. 1BestCsharp blog 3,713,963 views. Download with Google Download with Facebook or download with email. How to normalize and standardize time series data using scikit-learn in Python. 9 Exercises. RS – 4 – Multivariate Distributions 6 Example: The Multivariate Normal distribution Recall the univariate normal distribution 2 1 1 2 2 x fx e the bivariate normal distribution 1 2 2 21 2 2 2 1, 21 xxxxxxyy xxyy xy fxy e The k-variate Normal distributionis given by: 1 1 2 1 /2 1/2 1,, k 2 k fx x f e. One of the reasons is because when you use numpy you have to use numpy. How to do it The probability of a given value in a multivariate Gaussian distribution is calculated as follows: Import the relevant packages: from scipy. There IS a density with respect to the Hausdorff measure on the support, and this is exactly what stats. A Python implementation of Jerome Friedman's Multivariate Adaptive Regression Splines algorithm, in the style of scikit-learn. Multivariate. multivariate_normal function from numpy. Note that , and. I know it is a joke to do get it from there, but nevertheless. Numpy Multivariate Normal Distribution Anna Scaife University of Manchester Predicting the Future September 7, 2017 11 / 51 Motivation Gaussian Process Modelling CO2. 试图评估scipy的multivariate_normal. OK, I Understand. And covariance parameters, returning a frozen multivariate normal. multivariate_normalを使用して生成されます。 pyplot matplotlib. Take a sample of size 1,000,000 from a Normal with mean 0 and variance 1 and a sample of size 1,500,000 from a normal with mean 3 and variance 1. With earlier Numpy and Scipy versions, the results of such operations are undefined and usually unexpected. normal(): 正規分布（任意の平均、標準偏差） np. Computes the log of the probability density function of the normal N(mean, cov) for the data x. ndarray' object has no attribute 'pdf' Can anyone help me with what I am doing wrong. pdfs import Pdf you can type from pybayes import Pdf. The mean keyword specifies the mean.