Application of optical flow techniques to glial network Ca2+signalingMarius BuibasCSE252C FA2006 Project ReportAbstractThe use of the Optical Flow method is investigated for observation of intra-cellular Calciumsignaling in neural glia. Image sequences are captured for cells incubated with a Ca2+fluorescentdye, where increases in cytosolic calcium are indicated by an increase in image brightness. Afterthe cells are mechanically stimulated, they communicate with other connected cells, propagating thechange in calcium density and thus intensity. The optical flow technique is used to measure the speedof the signal and understand the cellular connectivity and receptor placement, by determining signalvelocities at all points in the image sequence. We find that the signaling behaviour and functionalconnectivity is highly complex, showing a wide range of signal propagation speeds and motifs.1. Background/MotivationThe study of complex networks is a central pursuit in a wide range of fields, from biology to engineer-ing to sociology, and determination of network connections for large (> 100) systems is a difficult ifnot intractable problem. With the recent development of Calcium sensitive dyes, signal propagationthrough a network can be observed by capturing image sequences using fluorescence microscopyfollowing cellular stimulation. In the Silva lab at the Jacobs Retina Center (silva.ucsd.edu), we aretrying to understand network dynamics and information flow in glial cellular networks [1]. Withinthe retina, glial cells function as helper cells to neurons, directing growth and removing by-products.Recently, glial cells have been found to communicate with one another and with neurons in sponta-neously growing networks [2]. Glial networks are of clinical importance, as they have been found tointerfere with neuron regeneration following injury.2. Optical Flow Method2.1. Theory and MethodsThe optical flow method is most commonly used in estimating object motion between subsequentframes. The method is based on the assumption that an object’s intensity does not change betweensubsequent frames, and that image pixel intensity changes between frames can be explained in termsof the movement of that particular object. Mathematically, the underlying assumption is that theintensity of a pixel I(x, y, t) on an object is constant over short time periods: (dIdt= 0). Expandingthis in an image space, with a moving pixel yields the common optical flow brightness constraintequation:∂I∂xdxdt+∂I∂ydydt+∂I∂t= 01Defining u =dxdtand v =dydtas the local velocity components, the optical flow constraint equationreduces to:∇I · (u, v) = Itimplying that a time change in intensity is the product of the local intensity gradient times the localvelocity. To solve for the local velocities, various constraints are proposed. Horn and Schunck [3]impose a velocity smoothness constraint, and develop an iterative method by minimizing deviationfrom the local velocity smoothness. A more effective method from Lucas Kanade [4] implements aweighted least-squares fit of local first-order constraints to a constant model of a local (u, v) in eachsmall spatial neighborhood Ω by minimization of the following expression:X(x,y)∈ΩW2(x, y) [∇I(x, y, t) · (u, v) + It(x, y, t)]2where W (x, y) is a window function that can be made to favor center constraints. The solution tothe above equation becomes:ATW2A(x, y) = ATW2bwhere, for n points (x, y) ∈ Ω, at a single time t,A = [∇I(x1, y1, t), · · · , ∇ I(xn, yn, t)]TW = diag [W (x1, y1, t), · · · , W (xn, yn, t)]b = − [It(x1, y1, t), · · · , It(xn, yn, t)]TThus the solution to the velocity vector u = (u, v) is:u = [ATW2A]−1ATW2bBarron, Fleet and Beauchemin [5], show improved performance of this method over the original Hornand Schunk iterative solution and was chosen here for analyzing glial signaling. The implementationdetails are outlined in the methods section.2.2. Application To Calcium SignalingWhile the original intent of the optical flow equation was to determine the velocity field for rigidobjects in image sequences, we have found the method to be useful in characterizing informationflow in glial networks, when observing image sequences of changing Ca2+concentrations in a fieldof cells following stimulation. Glial cells communicate intracellularly through external ATP diffusionand through gap junctions. As a result of this communication, Ca2+is released in the cytosol mainlyfrom sources in the endoplasmic recticulum (ER) inside the cell and can be observed with a Ca2+-sensitive dye. Ca2+is released quickly through channels that are activated and are afterwards pumpedout of the cytoplasm. It has been shown that Ca2+diffusion inside the cell is minimal and Ca2+stays local to the source before it is pumped back to the ER [6]. Ca2+does affect the release of othermessengers that propagate the signal to other areas of the cell and to other cells. Thus informationflow is defined as spatiotemporal progression of the cycle of Ca2+increase and decrease in visualfield. The direction and rate of propagation of the Ca2+information flow is the result of the complexdynamics of second messengers, and the non-homogenous location of receptors, gates, and pumpsinside the cell ultimately yield a moving Ca2+transient through the cellular network.2Astrocyte Movie Frame 114, along with line profile0 10 20 30 40 50 60050010001500200025003000350040004500Distance along line (pixelsLine Profile for 5 sequential framesInensity (12bit)f 114f 120f 126f 132f 138Figure 1: Line profiles for subsequent frames in selected region of astrocyte culture.When observing the traveling Ca2+transient, the rising edge will provide the highest spatiotem-poral derivatives and will most affect the local flow field calculation. Because the uptake of Ca2+is much slower than the release, the falling edge does produces a very small velocity component inthe calculation. This is illustrated in figure 1, where a series of intensity line profiles are shown forseveral frames, showing the progression of the Ca2+signal and subsequent activation of downstreamanother cell. Figure 2 shows the source frames along with computed optical flow at each frame ofthe same captured astrocyte culture sequence.One of the limitations of the optical flow method is the aperture problem, where the flow fieldestimation of large objects is incorrect inside the object, as there is no aparent intensity time change.In