The hypothalamic-pituitary-adrenal (HPA) axis is a major system maintaining body homeostasis by regulating the neuroendocrine and sympathetic nervous systems as well modulating immune function. Recent work has shown that the complex dynamics of this system accommodate several stable steady states, one of which corresponds to the hypocortisol state observed in patients with chronic fatigue syndrome (CFS). At present these dynamics are not formally considered in the development of treatment strategies. Here we use model-based predictive control (MPC) methodology to estimate robust treatment courses for displacing the HPA axis from an abnormal hypocortisol steady state back to a healthy cortisol level. This approach was applied to a recent model of HPA axis dynamics incorporating glucocorticoid receptor kinetics. A candidate treatment that displays robust properties in the face of significant biological variability and measurement uncertainty requires that cortisol be further suppressed for a short period until adrenocorticotropic hormone levels exceed 30% of baseline. Treatment may then be discontinued, and the HPA axis will naturally progress to a stable attractor defined by normal hormone levels. Suppression of biologically available cortisol may be achieved through the use of binding proteins such as CBG and certain metabolizing enzymes, thus offering possible avenues for deployment in a clinical setting. Treatment strategies can therefore be designed that maximally exploit system dynamics to provide a robust response to treatment and ensure a positive outcome over a wide range of conditions. Perhaps most importantly, a treatment course involving further reduction in cortisol, even transient, is quite counterintuitive and challenges the conventional strategy of supplementing cortisol levels, an approach based on steady-state reasoning. The hypothalamic-pituitary-adrenal (HPA) axis is one of the body's major control systems helping to regulate functions ranging from digestion to immune response to metabolism. Dysregulation of the HPA axis is associated with a number of neuroimmune disorders including chronic fatigue syndrome (CFS), depression, Gulf War illness (GWI), and posttraumatic stress disorder (PTSD). Objective diagnosis and targeted treatments of these disorders have proven challenging because they present no obvious lesion. However, the body's various components do not work in isolation, and it is important to consider exactly how their interactions might be altered by disease. Using a relatively simple mathematical description of the HPA axis, we show how the complex dynamical behavior of this system will readily accommodate multiple stable resting states, some of which may correspond to chronic loss of function. We propose that a well-directed push given at the right moment may encourage the axis to reset under its own volition. We use model-based predictive control theory to compute such a push. The result is counterintuitive and challenges the conventional time-invariant approach to disease and therapy. Indeed we demonstrate that in some cases it might be possible to exploit the natural dynamics of these physiological systems to stimulate recovery. The hypothalamic-pituitary-adrenal (HPA) axis constitutes one of the major peripheral outflow systems of the brain, serving to maintain body homeostasis by adapting the organism to changes in the external and internal environments. It does this by regulating the neuroendocrine and sympathetic nervous systems as well modulating immune function Many aspects of the organization and function of the HPA axis have been characterized in clinical and laboratory studies revealing a number of component feedback and feed forward signaling processes. Stress activates the release of corticotropin-releasing hormone (CRH) from the paraventricular nucleus (PVN) of the hypothalamus. The release of CRH into the hypophysial-portal circulation in turn acts in conjunction with arginine vasopressin on CRH-R1 receptors of the anterior pituitary stimulating the rapid release of adrenocorticotropic hormone (ACTH). ACTH then is released into the peripheral circulation and stimulates the release of the glucocorticoid cortisol from the adrenal cortex by acting on the receptor MC2-R (type 2 melanocortin receptor). Cortisol enters the cell and binds to the glucocorticoid receptor present in the cytoplasm of every nucleated cell; hence the widespread effects of glucocorticoids on practically every system of the body including endocrine, nervous, cardiovascular and immune systems.To keep HPA axis activity in check, glucocorticoids also exert negative feedback at the hypothalamus and pituitary glands to inhibit the synthesis and secretion of CRH and ACTH, respectively. Moreover, glucocorticoid negative feedback causes a reduction in corticotroph receptor expression leading to a desensitization of the pituitary to the stimulatory effects of CRH on ACTH release. This negative feedback is also felt in the hippocampus where it exerts a negative influence on the PVN. A detailed review of the physiology and biochemistry of the HPA axis as well as it's know interactions with the immune system may be found in work by Silverman et al. A number of chronic diseases have been characterized by abnormalities in HPA axis regulation. These include major depression and its subtypes, anxiety disorders such as post-traumatic stress disorder, panic disorder and cognitive disorders such as Alzheimer's disease and minimal cognitive impairment of aging Though much is known about its components, one of the main difficulties in studying the behavior of the HPA axis has been in integrating the expansive body of published experimental information. Numerical models provide an ideal framework for such integration. Simple models of the HPA axis have been constructed using deterministic coupled ordinary differential equations In this work we adopt the model proposed by Gupta et al. H (Eq. 1).A model of the HPA axis which includes glucocorticoid receptor and the dynamics of glucocorticoid receptor-cortisol interactions have been proposed by Gupta et al. 1; x2; x3; x4]T and are described in i1; kcd; kad; ki2; kcr; krd; k]T. Nominal values for the system parameters are listed in H from a steady-state value.The system states are given as x = [xIn this first analysis the HPA axis system is considered under idealized conditions where all parameters are assumed constant and precisely known. In addition, the states x are assumed known as a function of time with no measurement error and the control action is implemented perfectly. The approach taken for choosing an optimal control is based on the Model Predictive Control (MPC) framework H is augmented with a control term u in the equation for cortisol (x4) (Eq. 2). Note that System Hu is affine with respect to the control action and the disturbance.In this work it is assumed that the variable to be manipulated for treatment is the rate of addition or removal of cortisol from circulation. To model this control action, System u(t) we define the following penalty function to enforce minimal departure from normal ACTH (x2) and cortisol (x4) levels even though we purposely manipulate circulating cortisol to perturb the system uH.To avoid dangerous destabilization of the HPA axis by the application of control action 0 and tf are the start and end time of the optimization horizon, λ is a tuning parameter taking values from zero to one and x2* and x4* are the healthy steady-state concentrations of ACTH and cortisol, respectively. R is a penalty assigned to the input and Q is the penalty assigned to the state variables. R was chosen as 0 because the cost for therapy was considered negligible compared to the cost of ongoing disability. Q was chosen as follows because x2 and x4 are the only measured states.Where tThe resulting cost function can be written as:1, x2, x3) in response to the control action applied to cortisol (x4). In this case, the objective of the controller is to bring the cortisol concentration to set point while minimizing the impact of the treatment on the other three states of the HPA axis. Any change in CRH (x1) or the glucocorticoid receptor  will be reflected in the concentration of ACTH (x2) by virtue of the coupled dynamics described by System Hu. The tuning parameter λ can be selected to match the intensity of the desired treatment. A λ value of near zero will lead to a more intense treatment while a value of λ near one will lead to very conservative treatment. For proof of concept, a more direct treatment was favored in this work and a λ value of 0.01 was used throughout. Note that x2* and x4* correspond to the stable steady state of the unperturbed system . As a result, once the system has been brought to the healthy steady state it will stay at this steady state even if the external control action (treatment) is removed.The parameter λ is used to penalize excessive imbalance of the other hormones (x4(u(t))x. That is, the optimization procedure searched for an optimal input in the set Uc of all piecewise constant functions on Typically, a treatment or control action is applied at discrete intervals. As a result, the objective function in Equation 3 was optimized with respect to a piece-wise constant input signal 0) is the steady state of the unperturbed system with d0 = 0. The optimal control, The initial condition, x. A plot of the steady-state values of x1, x2 and x3 as a function of d0 is shown in Therein f0 = 0. If a healthy person were subjected to extreme stress  for an extended period of time their body would reach the only steady state available locally that is one corresponding to chronic stress. In other words, for values of d0 greater than 0.168, Equation (9) dictates that there is only one steady-state solution for free GR (x3) concentration as opposed to the 3 solutions available for 0≤d0<0.168. By virtue of Equation (12) this results in only one steady-state solution being available for cortisol (x4) for d0>0.168. When the stress is removed , the body will stay at this new depressed steady-state value of cortisol concentration. This process is shown graphically in In this model of HPA axis dynamics a chronically stressed individual would occupy the stable steady state associated with a depressed cortisol concentration (∼0.05) at rest or at dAs one might expect the assumptions of ideal control do not correspond to a physically realizable system. However, the analysis of the system under idealized conditions allows the study of possible treatments. Any practical treatment would then be a suboptimal solution as compared to the treatment under idealized conditions. This allows proposed treatments to be benchmarked and compared. In addition, the solution obtained under idealized conditions can serve as a qualitative guideline for the creation of a practical, although suboptimal, treatment. In engineering terms the objective of treatment is to succeed in bring the subject to the healthy steady-state target while exerting the smallest disturbance possible to the HPA axis. For example, even though we intend to manipulate circulating cortisol concentration it should not be allowed to decrease excessively because of the important role cortisol plays in regulating a number of cellular and physiological functions. To avoid such excess perturbations the concentration of ACTH has been included in the objective function of Equation 3.This concentration is more readily measured than that of either CRH or GR making ACTH a good candidate for monitoring the progress of a treatment.The optimal control solution that minimizes disruption of HPA axis function (Eq. 3) is shown in 2 and x4, respectively, can be readily measured. The availability of cortisol analogues makes it possible to manipulate x4 directly. Therefore as postulated previously (Eq. 3–4) ACTH (x2) can be used to determine when a change in available steady state or attractor has occurred. Under the MPC framework, most of the control action is expended near the initial time. In 2) are both plotted as a function of time. The value of x2 increases by about 30% as the system moves from the cusp of multiple candidate steady states to the basin of a single steady state. The following treatment is therefore proposed:In this section a suboptimal control strategy is proposed for the HPA axis system. The goal of this strategy is to mimic the qualitative results of the MPC solution while being realizable in a clinical setting. The MPC solution suggests that manipulating cortisol concentration is a plausible strategy for redirecting the HPA axis to a healthy steady state. The key difficulty in applying this approach is determining when the cortisol concentration has been sufficiently lowered with regard to the other state variables to allow the system to return to a healthy equilibrium. That is, one must identify an observable event (corresponding to a measurable variable) which signals that the steady state of the system has shifted. In a clinical setting only ACTH and cortisol concentrations, corresponding to xTreatment 1 The cortisol concentration in the system should be slowly decreased until ACTH levels (x2) have increased by more than 30% relative to the initial condition. Once this signal is observed, the system's own natural feedback control action should restore cortisol levels to normal.Simulation results for Treatment 1 are shown in 0) is examined in this section. A direct computational evaluation of robustness of Treatment 1 is difficult to implement. There are four initial conditions (x1(0); x2(0); x3(0); x4(0)), seven parameters, and one disturbance variable (d0). A simulation study where each variable  is evaluated at a nominal, high and low values, would require, at a minimum 312 = 531,441 simulations. Even if these simulations were completed, the choice of high, low and nominal value for each variable would be difficult to justify using available data. An alternative approach analyzing robustness analysis is to study the asymptotic behavior of System uH. Let the concentration of cortisol (x4) be manipulated so that the product of cortisiol and GR concentrations (x3x4) is constant. Under these conditions, the asymptotic value of glucocorticoid receptor concentration GR (x3) is obtained from Eq. 7 as:The results for Treatment 1 shown in 3∞ has a minimum as a function of cortisol concentration (x4) at x4 = 0. That is, if one were to lower the cortisol concentration to zero one would obtain the lowest possible steady-state value for GR and this value would be:The asymptotic value or GR concentration x4∞ = 0 and x3∞ = kcr/krd the unique asymptotic solution for CRH (x1) and ACTH (x2) is given byAt the steady-state point given by xuH). For the closed loop HPA axis system it represents the minimum achievable cortisol concentration. Note that this equilibrium point is only achievable under external input. This result is independent of the trajectory of the input u(t) and is a property of the HPA axis. Moreover the solution in Equation 15 is unique indicating that only a single steady state exists at the minimum for x4→0 and that this state corresponds to a stable set of non-zero real-valued concentrations of CRH, ACTH and GR. This result confirms that reducing the cortisol concentration to a small enough positive value can indeed take the system to a single stable condition. This is regardless of the value of d0, parameters, or initial conditions. This condition will correspond to a healthy equilibrium value when treatment is administered in the absence of elevated levels of external stress d0. At high levels of the stressor d0 the success of the treatment would be short lived as we would simply be immediately re-administering the same insult originally responsible for the illness state. This is true regardless of whether the idealized or the suboptimal treatment approach is used.It should be noted that the values for GR, CRH and ACTH identified in Equations 14–15 represent an asymptotic minimum for the externally controlled system  inhibition of cortisol synthesis at the level of the adrenal gland or (2) inhibition of CRH induced ACTH synthesis by the pituitary. Inhibitors of cortisol synthesis include pharmaceutical agents such as ketoconazole that have been used in limited human trials Indirect approaches to cortisol suppression focus on modulation of the biochemical feedback returning to the higher HPA axis from the immune system and the adrenal gland. Inflammatory events exert a positive immune system feedback to the HPA axis that is conducted via a number of pro-inflammatory cytokines for which several components of the HPA axis have receptors. Supported by immune, epidemiological and small-scale gene expression data Unfortunately as in strategies involving the direct inhibition of CRH, a reduction of positive feedback to the hypothalamus also leads to a reduction in ACTH synthesis by the pituitary. Recall that the proposed treatment requires the inhibition of the negative cortisol feedback without the removal of positive stimulation of ACTH production. This could be achieved by temporarily reducing the bioavailability of cortisol itself. Binding proteins and metabolizing enzymes have been identified for cortisol. Corticosteroid-binding globulin (CBG) regulates the concentration of free or active cortisol It should be noted that although the model of HPA axis dynamics used in this work is currently the most credible model, it remains in many ways incomplete. For example, there is mounting data including observations of moderate hypocortisolism in depressed patients undergoing IFN-α therapy suggesting that GR receptor function not only affects the release of cytokines but is itself affected by these same cytokines It is important to note however that while the specific treatment solution identified using MPC is model-dependent the general MPC framework is not. Therefore as more detailed models become available these can easily be exploited to improve a treatment course. Putting aside issues of model fidelity and completeness, the proposed MPC framework could still be exploited in a two-step treatment approach. In a first step data obtained from a standard dexamethasone test could serve to calibrate a simple lumped-parameter model capturing the overall HPA dynamics for a given subject. The calibrated model could then be used within the proposed MPC framework to estimate the most appropriate combination of dosage and duration of treatment for that same patient. Ultimately even if a given model is not entirely correct our robustness analysis shows that the desired outcome may be obtained reliably over a wide range of parameter values. This will be true as long as the structure of the model is valid.In conclusion we have demonstrated in this work the use of model-based predictive control methodology in the estimation of robust treatment courses for displacing the HPA axis from an abnormal hypocortisol steady state back to a normal function. Using this approach on a numerical model of the HPA axis proposed by Gupta et al.


Flat panel detector-based cone beam breast CT (CBBCT) can provide 3D image of the scanned breast with 3D isotropic spatial resolution, overcoming the disadvantage of the structure superimposition associated with X-ray projection mammography. It is very difficult for Mammography to detect a small carcinoma (a few millimeters in size) when the tumor is occult or in dense breast. CBBCT featured with circular scan might be the most desirable mode in breast imaging due to its simple geometrical configuration and potential applications in functional imaging. An inherited large cone angle in CBBCT, however, will yield artifacts in the reconstruction images when only a single circular scan is employed. These artifacts usually manifest themselves as density drop and object geometrical distortion that are more noticeable in the reconstructed image areas that are further away from the circular scanning plane. In order to combat this drawback, a circle plus partial helical scan scheme is proposed. An exact circle plus straight line scan scheme is also conducted in computer simulation for the purpose of comparison. Computer simulations using a numerical breast phantom demonstrated the practical feasibility of this new scheme and correction to those artifacts to a certain degree. Breast cancer imaging has improved over the last decade with higher and more uniform quality standards for mammography as well as through the increasing use of sonography and magnetic resonance imaging as the adjunct tools. Mammography is still the only screening tool to detect breast cancer for asymptomatic women. Due to the limitations associated with the aforementioned techniques, such as imaging of the overlapping structure with mammography, technician dependent lack of ability to detect calcifications with ultrasound, and low specificity and/or poor detection of the tiny calcium deposits with MRI, there remains an endeavor to explore new ways to better detect breast cancer. Recently one of the most exciting ways is cone beam breast CT (CBBCT) technology –4. It isAmong all CBBCT technologies, FDK , 8 algorRecently, Katsevich and Kapralov  proposedXOY defines a scanning plane, and the point C represents one X-ray source on the circle scan trajectory; O is the rotation center; OC is the diameter by which a Radon shell is defined; Do is a point on the Radon shell; N is a point where the line CDo intersects the virtual detector. C1C2 is a line that crosses the point N and is perpendicular to the line ON, CC1C2 defines a plane  where its normal is O to this plane which is also the length of the line ODo is ρ. The corresponding Radon point Dr in Radon domain that is defined by the Radon plane CC1C2 in the object space is illustrated in Z axis) to constitute a torus in a 3-D Radon domain. In the CBBCT scanning geometry, the aforementioned Radon shell becomes a half Radon shell on the scanning plane; so only a half Radon ball is shown.It is well known that a single circular cone beam scan does not provide complete information for an exact reconstruction. This can be appreciated by the 3-D Radon transform of the object functionρ > OC · sinθ. We can make two claims by observing this missing Radon volume. (1) When the sampling rate is fixed, more Radon points are needed to fill this missing volume in the part further away from the scanning plane than in the part closer to the scanning plane. This actually indicates that the reconstruction based on the circular scan has more artifacts in the reconstructed slices that further away from the scanning plane than those closer to the scanning plane. (2) The ratio of the radius of the object support and the diameter of the spherical Radon shell determines the size of the missed Radon data volume, which results in the reconstructed object that is closer to or farther away from the exact reconstruction. With a fixed diameter  of the Radon shell, it is evident that the smaller the breast, the better the reconstructed image; the bigger the breast, the worse the reconstructed image in terms of artifacts.The light gray volume inside the half Radon ball support is what is called the missing volume, meaning that no Radon points in this volume can be acquired through circular scan. In the spherical coordinates, this missing Radon volume is expressed as According to Chen and Ning , when thϕL(l) = , where m is a constant in the Y-axis, and l is a variable along the Z-axis, representing the line scan X-ray shot position, based on the illustration from XOZ plane. In XOZ plane. Since the missing Radon volume is symmetrical around the rotation axis (Z axis), then this unsymmetrical filling of the Radon data in terms of projection angle in the missing Radon volume may not achieve the best reconstruction result. By taking advantage of the circular scanning feature of CBBCT, one way to combat this unsymmetrical filling is to lower down the X-ray tube and detector while simultaneously rotating them around the breast to achieve an approximate symmetrical filling of the Radon data in the missing Radon volume. It is like the helical scan but with sparse X-ray shots at positions described as ϕHL = , where D is the distance from the X-ray tube to the rotation center, and li is the position along the Z axis, and can be described as li = l0 + (i − 1)Δl, where l0  is the starting position along the Z axis for this partial helical scan, i is index of X-ray shot, Δl is the line increment along the Z axis, and β is the projection angle, and also can be described as βi = (i − 1)Δβ, where Δβ is the projection angle increment in the unit of radians. The Radon data acquired through this scanning trajectory can fill the part of this missing Radon volume. Thus the result is not an exact reconstruction. The key point here is to introduce the additional scanning trajectory so as to correct to a certain degree the reconstruction artifacts associated with a single circular scan.There are a couple of proposed “circle plus” trajectories , 15–18. For comparison, a CL scanning is also conducted in numerical simulation based on Katsevich's concept under the less restrictive conditions. During the line scanning, the detector is always fixed at the position where circle scan is conducted, and the X-ray collimation size varied to make sure that X-ray illumination always covers the whole detector as it moves along the line trajectory. In this way, the missing radon volume is filled completely and an exact reconstruction can be achieved through CL scan.z = 0 cm during the circular scan. After the circular scan, the X-ray source and detector lower down simultaneously while they are still rotating. When the X-ray source gets to the point where z = l0 , it starts to shoot at positions described as ϕHL =  (i = 0 to n), and the X-ray source maintains the same half cone illumination for each shooting as it is with circular scan; so part of the breast in each helical shot between z = 0 and z = il  can avoid being exposed by the X-ray. Based on the geometric parameters of current CBBCT, we designed a new scan scheme. The position of the X-ray source is at π range. There are32 and 64X-ray shots during partial helical scans that uniformly cover the angular range of 2π, and the movement in the Z direction is from 49 mm to 121 mm with the increment interval of 2.34 and 1.15 mm based on the size of the simulated breast phantom. The reason we chose the starting position at Z = 49 mm for partial helical scan is because we found that based on our simulated scanning geometrical parameter the attenuation coefficient drop in the regular circular scan started approximately at Z = 49 mm.The projection angles associated with partial helical scans are uniformly distributed within 2Some of the Radon data points acquired from this additional scanning trajectory still can be acquired through a circular scan. This is what is called redundant sampling points in the Radon domain and can be efficiently eliminated by a window function. The geometric setup of the collimation during the partial helical scan is maintained as it is with the circle scan, that is, the half cone illumination geometry. This can avoid the redundant sampling in the missing volume in the Radon domain within the X-ray shots in a helical trajectory. Since the collimation during partial helical scan unavoidably encounters the longitudinal truncation, a geometric dependent truncation window function has to be used to handle this case to remove the incorrect Radon data.W. Those little black dots represent the X-ray source at different positions in line scan. During the line scan, the detector is fixed and the collimation of the X-ray is adaptively changed to cover the whole detector, and the length of this scanning line is 2W. The circle trajectory can always be projected onto the detector as it is seen from the X-ray source during the straight line scan, thus enabling us to use Katsevich's algorithm to do the reconstruction for this line scan part.In line scan case, as Composite reconstruction framework is probably the most preferable algorithm for the CBBCT. The reconstructed object isThe mathematic equation of essed by  and 5),,fcir(r⃗)(i) FDK algorithm:h(t) is the impulse response of the regularized ramp filter; Pβ is the cone beam projection data.(ii) Hui's term:wZi is used to remove Radon points that are acquired through partial helical scan but have already been touched by previous circular scan during the reconstruction. As SC1C2 defined in the reconstruction coordinates during partial helical scan corresponds to a Radon point expressed as  in terms of spherical coordinates. This Radon point must be mapped to the Radon domain defined by the object coordinates expressed as  in order to construct the window function wZi.  PZi is the projection data associated with each X-ray position during partial helical scan.Based on essed as . As was wtrZi is included in this paper. Based on the scanning design, the partial helical scan will unavoidably encounter the longitudinal truncation during the scan. Some Radon points it acquires do not reflect the actual Radon data and should be removed during the back-projection  except Figures The simulation was conducted in several settings as discussed in Section The contribution from straight line scan was reconstructed using Katsevich's algorithm.RE) is conducted according to the following formula:A quantitative measurement of reconstruction error  is the CT number of the ith pixel in the reconstructed central sagittal image; Ip(i) is the CT number of the same pixel in the central sagittal phantom image.RE from different scan schemes.7 photons/cm2 · mR [In order to test the performance of this new scheme over the quantum noise that is commonly encountered in practical CBBCT data acquisition, we generated quantum noise contaminated data. An X-ray with 60 kVp was selected which corresponds to an effective photon fluence of 2.65*10cm2 · mR . The expThe new scanning scheme of CH scan works better than a single circular scan in terms of image uniformity and geometrical correctness based on the computer simulations of a mathematic breast phantom and a simulated breast phantom on CBBCT prototype study. Partial helical scan with different sampling intervals showed that the number of X-ray shootings between 32 and 64 could provide acceptable reconstructed images in terms of correction to the intensity drop along the scanning axis and geometrical distortion around the nipple area based on the scanning geometrical parameters and breast size. This is encouraging, since the quality of reconstructed images could improve without too much additional radiation exposure to the patient. Also note that the smaller the sampling interval (the larger the number of projections) in helical scan, the less the streak artifacts in the corrected area. However, these streak artifacts are faintly visible. In practical situation, the image quality should be balanced with the sampling interval in helical scan. This new scanning scheme is not intended to conduct an exact reconstruction. Theoretically, when the missing volume in Radon domain is completely filled and at least as densely sampled as those accessed by circle scan, the combined reconstruction is exact. By sparsely sampling the missing volume through a proposed scanning scheme, it suffices to correct the artifacts occurring in a single circular scan. On the other hand, the new scanning scheme is easy to operate in practice without complicated mechanical modification on the current prototype CBBCT system. As was mentioned in Visually, the reconstruction from CL scheme looks smoother than that from CH scheme; all the streak artifacts noticed in CH are gone in CL. This can also be appreciated from the profile comparison. The profile comparison in In conclusion, by incorporating a sparse partial helical scanning trajectory into an FDK-based single circular scanning scheme, a new circle plus partial helical scanning scheme was proposed to compensate for the artifacts inherited by a single circular scan for CBBCT prototype system. The numerical simulation study has demonstrated its feasibility.


Sir,1 that the CD1a expression by dendritic cells (DC) in human cancers is a very interesting observation and that the possible role of the CD1a molecule in the presentation of tumour-associated antigens requires further investigation. We were intrigued to note in the Cappello et al paper2 that CD1a was expressed in 75–94% of epithelial biopsy tissue sections from patients with Barrett's metaplasia of the oesophagus, but that this was essentially not found in patients with dysplasia or neoplasia, nor in normal gastric or colonic tissues examined. Although the study was small, the report brief, and did not contain more advanced invasive carcinomas, it might indicate that transitory expression of CD1a by epithelia may occur during the initial phases of carcinogenesis. We have some unsubstantiated data that also indicate that some in situ breast carcinomas may also show expression of CD1a. It is unclear whether the expression of the CD1a molecules is actually on the epithelial cells per se or on intraepithelial DC. We have theorised that if CD1a expression occurs on both epithelial cells and DC, then this may indicate a similar arrangement to that observed for MHC molecular expression for presentation of peptide antigens. This would strengthen our hypothesis that CD1a, with some known structural similarity to MHC, may be very important for the presentation of tumour-derived glycolipid antigens to T cells. In addition, we are currently investigating CD1a restricted T-cell responses against tumour antigens, and we hypothesise that these are important for effective antitumour immune responses in vivo. If epithelial cells in transition towards neoplasia express CD1a together with neo-antigens, this may be a sufficient ‘danger’ signal to alert the immune system to early carcinogenesis, thus inducing an effective immune response in some individuals – perhaps as a mechanism for epithelial protection against further malignant progression. The failure of expression of CD1a by epithelia in transition may not allow an immune response to occur, perhaps explaining the worse prognosis of CD1a-negative oesophageal lesions noted in the study of Cappello et al. This theory remains to be (dis)proved.We agree with Dr Cappello's comment relating to our paper


Specifically, φ quantifies how much information is generated (uncertainty is reduced) when a system enters a particular state through causal interactions among its elements, above and beyond the information generated independently by its parts. Such mathematical characterization is motivated by the observation that integrated information captures two key phenomenological properties of consciousness: (i) there is a large repertoire of conscious experiences so that, when one particular experience occurs, it generates a large amount of information by ruling out all the others; and (ii) this information is integrated, in that each experience appears as a whole that cannot be decomposed into independent parts. This paper extends previous work on stationary systems and applies integrated information to discrete networks as a function of their dynamics and causal architecture. An analysis of basic examples indicates the following: (i) φ varies depending on the state entered by a network, being higher if active and inactive elements are balanced and lower if the network is inactive or hyperactive. (ii) φ varies for systems with identical or similar surface dynamics depending on the underlying causal architecture, being low for systems that merely copy or replay activity states. (iii) φ varies as a function of network architecture. High φ values can be obtained by architectures that conjoin functional specialization with functional integration. Strictly modular and homogeneous systems cannot generate high φ because the former lack integration, whereas the latter lack information. Feedforward and lattice architectures are capable of generating high φ but are inefficient. (iv) In Hopfield networks, φ is low for attractor states and neutral states, but increases if the networks are optimized to achieve tension between local and global interactions. These basic examples appear to match well against neurobiological evidence concerning the neural substrates of consciousness. More generally, φ appears to be a useful metric to characterize the capacity of any physical system to integrate information.This paper introduces a time- and state-dependent measure of integrated information,  integrated information. This suggestion stems from considering two basic properties of consciousness: (i) each conscious experience generates a large amount of information, by ruling out alternative experiences; and (ii) the information is integrated, meaning that it cannot be decomposed into independent parts. We introduce a measure that quantifies how much integrated information is generated by a discrete dynamical system in the process of transitioning from one state to the next. The measure captures the information generated by the causal interactions among the elements of the system, above and beyond the information generated independently by its parts. We present numerical analyses of basic examples, which match well against neurobiological evidence concerning the neural substrates of consciousness. The framework establishes an observer-independent view of information by taking an intrinsic perspective on interactions.We have suggested that consciousness has to do with a system's capacity to generate  Scientists and engineers are usually interested in how information can be transmitted or stored from the perspective of a user. However, it is just as important to consider information, in the classic sense of reduction of uncertainty, from the perspective of an autonomous system. How much information is generated when the system enters a particular state by virtue of causal interactions among its elements? And to what extent is the information generated by the system as a whole, as opposed to the information generated independently by its parts? Addressing these questions requires the development of a new framework that is based on the notion of integrated information The need for such a framework is not merely academic. Indeed, it was initially motivated by one of the most baffling scientific problems – the generation of conscious experience by the brain. We know that certain regions of the brain, for example the thalamocortical system The integrated information theory of consciousness represents an attempt to address the first problem of consciousness from first principles In the present work, our goal is to provide a definition and measure of integrated information for systems of discrete elements that evolve through time. This extension provides a framework for integrated information that is fully general and can be applied in principle to any kind of physical system. It also permits further predictions concerning the relationships between brain processes and consciousness. Finally, irrespective of the relevance for understanding consciousness, the notion of integrated information presented here may be useful for characterizing computational systems not merely as processors or stores of information, but as integrators of information.i) there is a large repertoire of conscious experiences. This means that, when one particular experience occurs, it generates a lot of information; ii) each experience is integrated, i.e. it appears as a whole that cannot be decomposed into independent parts It is useful to briefly examine the phenomenological observations that motivate the integrated information approach to consciousness. From a first-person perspective – the perspective of the system that is actually capable of generating subjective experience – two fundamental properties of consciousness are apparent: Consider the following: You are facing a blank screen that is alternately on and off, and you have been instructed to say “light” when the screen turns on and “dark” when it turns off. A photodiode – a very simple light-sensitive device – has also been placed in front of the screen, and is set up to beep when the screen emits light and to stay silent when it does not. The first problem of consciousness reduces to this: when you distinguish between the screen being on or off, you have the subjective experience of seeing light or dark. The photodiode can also distinguish between the screen being on or off, but presumably it does not have a subjective experience of light and dark. What is the key difference between you and the photodiode?According to the theory, the difference has to do with how much information is generated when that distinction is made. Information is classically defined as reduction of uncertainty when a particular outcome occurs out of a repertoire of alternative outcomes: the more numerous the outcomes, the greater the reduction of uncertainty, and thus the information. When the blank screen turns off, the photodiode enters one of its two possible states and beeps, yielding 1 bit of information. However, when you see the blank screen turn off, the state you enter rules out a very large number of possible states. Imagine that, instead of turning homogeneously off, the screen were to display at random every frame from every movie that was ever produced. Without any effort, each of these frames would cause you to enter a different state and see a different image. This means that when you enter the particular state (“seeing pure darkness”) you rule out not just “seeing light,” but an extraordinarily large number of alternative possibilities. Whether or not you think of the bewildering number of alternatives (you won't and you can't), this corresponds to an extraordinary amount of information. Importantly, this information has nothing to do with how complicated the scene is – pure darkness or a busy city street – but only with the number of alternative outcomes.1,000,000 states, an immense number, corresponding to 1,000,000 bits of information. Indeed, the camera would enter a different state for every frame from every movie that was ever produced. Yet few would argue that the camera is conscious. What is the key difference between you and the camera?While the ability to distinguish among a large number of states is a fundamental difference between you and the photodiode, by itself it is not enough to account for the presence of consciousness. To see why, consider an idealized megapixel digital camera, whose sensor chip is essentially a collection of a million photodiodes. Even if each photodiode in the sensor chip were just binary, the camera could distinguish among 21,000,000 states. In reality, however, the chip is not an integrated entity: since its 1,000,000 photodiodes have no way to interact, the state of each photodiode is causally independent of that of the others: in reality, the chip is a collection of 1,000,000 independent photodiodes, each with a repertoire of 2 states. This is easy to prove: if the sensor chip were cut down into its individual photodiodes, the performance of the camera would not change at all. By contrast, your vast repertoire of conscious states truly belongs to an integrated system, since it cannot be subdivided into repertoires of states available to independent components. Thus, a conscious image is always experienced as an integrated whole: no matter how hard you try, you cannot experience the left half of the visual field of view independently of the right half, or colors independent of shapes. Underlying this unity of experience are causal interactions within your brain, which make the state of each element causally dependent on that of other elements. Indeed, unlike the camera, your brain's performance breaks down if its elements are disconnected. And so does consciousness: for example, splitting the brain in two along the corpus callosum prevents causal interactions between the two hemispheres and splits experience in two – the right half of the visual field is experienced independently of the left.According to the theory, the difference has to do with integrated information. An external observer may consider the camera chip as a single system with a repertoire of 2and it must be unified, i.e. it should not be decomposable into a collection of causally independent subsystems (integration). How can one establish the size of the repertoire of states available to a unified system?This phenomenological analysis suggests that, to generate consciousness, a physical system must have a large repertoire of available states (information) Our goal is to provide a way to measure how much information is generated when a physical system enters one particular state out of a repertoire of possible states, but only to the extent that the information is generated by the system as a whole, above and beyond the information generated independently by its parts. Previous work To evaluate how much integrated information is generated when a system enters a particular state, we consider simple systems composed of a few interacting elements. Though the present framework is meant to be general, it is convenient to think of neural elements that can be active (fire) or inactive and can communicate through directed connections.X be a system consisting of n elements, which are taken to be abstract indivisible units. Each element is assumed to have a finite repertoire of outputs, with no accessible internal structure. In the examples below the repertoire of the elements will typically consist of two outputs: 0 or 1, corresponding to silence or firing. The internal states of the elements are irrelevant because it is only through outputs that an element can causally affect the rest of the system.Let AND, XOR; they can also be probabilistic.Elements are linked by connections to form a directed graph, specifying which source elements are capable of affecting which target elements. Each target element is endowed with a “mechanism” or rule through which it determines its next output based on the inputs it receives. These mechanisms are assumed to be elementary, for example t depends only on the inputs at time t−1. In future work we will extend the framework to include elements with memory and explain how the natural time frame over which a system generates integrated information is specified.Time is assumed to pass in discrete instants, which could correspond to milliseconds for example. We use the word state to refer to the total output of a given subset of a discrete system at a given instant in time. Finally, the elements are memoryless, meaning they are modeled as first order Markov processes: the output of an element at time X, S and so forth. Uppercase letters with subscripts  denote probability distributions of perturbations that are physically imposed on the outputs of a subset at a given time, e.g. at t = 0. Lowercase letters with subscripts  denote events: the actual output of the subset in question at a particular time, e.g. at t = 1.We refer to systems and subsets of systems by capital letters: x1, out of its repertoire (a repertoire is a probability distribution on the set of output states of a system). The information generated should be a function of how large the repertoire of possible states is, and how much uncertainty about the repertoire is reduced by entering state x1. Also, the reduction of uncertainty must be produced by interactions among the elements of the system acting through their causal mechanisms, which is why we call it effective information.First, we need to evaluate how much information is generated by a system when it enters a particular state, AND-gates and transitions from state x0 = 110 at time zero to state x1 = 001 at time one. How much effective information does the system generate? To answer the question we need to precisely describe: i) the alternative states available to the system (the a priori repertoire); ii) those states that the architecture of the system specifies as causes of x1 (the a posteriori repertoire). Effective information captures the information generated by the system by measuring the difference between these two repertoires.Let us first consider an isolated system, as in Effective information is defined as the entropy of the a posteriori repertoire relative to the a priori repertoire, which we write as:The a priori repertoire is the probability distribution on the set of possible outputs of the elements considered independently, with each output equally likely. This repertoire includes all possible states of the system prior to considering the effects of its causal architecture and the fact that it entered state x1. This distribution is imposed onto the system, i.e. we perform a perturbation in the sense of a priori repertoire coincides with the maximum entropy (maxent) distribution on the states of the system; we denote it by pmax(X0). No perturbation can be ruled out a priori, since it is only by passing a state through the mechanism that the system generates information. The maximum entropy distribution formalizes the notion of complete ignorance a priori repertoire distribution assigns equal probability to each of the 23 = 8 possible outputs of the system.The a posteriori repertoire p(X0 → x1) is the repertoire of states that could have led to x1 through causal interactions. We determine the a posteriori repertoire by forcibly intervening in the system and imposing each state in the a priori repertoire, thus we implement a perturbational approach a priori perturbation in turn we find that some perturbations could have caused (led to) x1 and others not . The a posteriori repertoire is formally captured by Bayes' rule, which keeps track of which perturbations cause (lead to) the given effect )is the entropy of probability distribution p. Entropy of the a priori repertoire n bits in a system of n binary elements. The second term is the entropy of the a posteriori repertoire, and lies between 0 and n bits depending on the state x1 and the architecture of the system. It follows that a system of n binary elements generates at most n bits of information.Here a priori repertoire is 3 bits and that of the a posteriori is 0 bits, so 3 bits of effective information are generated by the system when it enters x1: one out of eight perturbations is specified by the system as a cause of its current state, and the other 7 perturbations are ruled out, thus reducing uncertainty (generating information).In a priori repertoire is smaller than in AND-gate system entering state x1 = 000. In this case the a posteriori repertoire specified by the system contains four perturbations that cannot be distinguished by its causal architecture, since each of the four perturbations leads to 000. Fewer alternatives from the a priori repertoire are ruled out, so effective information is 1 bit.In x1 = 111. The process of entering x1 does not rule out any alternative states, so the a posteriori repertoire coincides with the a priori repertoire and effective information is zero. In x1 does not rule out any alternative states, so the a posteriori repertoire coincides with the a priori repertoire and effective information is zero.Finally, X in some larger system W that forms the “world” of X. Inputs from the environment, E = W \ X, to X cannot be accounted for by X internally. From X's point of view they are a source of extrinsic noise, since the information generated by the system must be due to causal interactions within the system. In general, to compute effective information one should average over all possible external inputs with the maximum entropy distribution  generated by the system, considered as a single entity, is 4 bits. In this case, however, it is clear that the two couples do not constitute a single entity at all: since there are no causal interactions between them, each of the disjoint couples generates 2 bits of information independently  is defined as the entropy of the a posteriori repertoire of the system relative to the combined a posteriori repertoires of the parts:μ stand for parts, and MIPP is the minimum information partition, which represents the natural decomposition of the system into parts.The a posteriori repertoires of the parts are found by considering each part as a system in its own right  by its causal architecture and current state a priori repertoires relative to their a posteriori repertoires.a posteriori repertoire) with respect to (over and above) the information generated independently by its parts (their combined a posteriori repertoires). In particular, integrated information is zero if and only if the system can be decomposed into a collection of independent parts. Thus, φ(x1) of a system captures how much “the whole is more than the sum (or rather the product) of its parts.”Integrated information, shown in dark blue, measures the information generated by the system through causal interactions among its elements , coinciding with the a posteriori perturbation specified by the entire system. No alternatives are ruled out by the system as a whole, so integrated information isTo exemplify, consider again the system of The system generates no information as a whole, over and above that generated by its parts.Of note, a related measure is stochastic interaction MIPP. Considering other partitions, for example partition MIPP, which can be found by searching over all partitions of the system after appropriate normalization.In the case of the two couples the natural decomposition of the system into parts is captured by partition effective information across an arbitrary partitionP = {X}. Since the part is the entire system, the a posteriori repertoire of the part equals that of the system, so if we apply Equation (2A) to the total partition we always obtain zero. Thus we define effective information across the total partition to be effective information generated by the entire system, as in Equation 1A. For a system containing no elements with self-connections effective information generated by the system  and effective information across the partition into individual elements coincide.To do so, let us define the m is the number of parts in the partition. The normalization is the size of the smallest a priori repertoire of a part multiplied by the number of other parts. In particular, for a partition into two parts PN is the size of the smaller a priori repertoire. The normalization for the total partition is PN = Hmax(X0).Normalization is necessary because effective information across an asymmetric bipartition where one part contains a single element and the second part contains the rest will typically be less than across a symmetric partition into two parts of equal size. Similarly, effective information across partitions into many parts tends to be higher than across partitions into few parts. To fairly compare different partitions we therefore introduce the normalization:minimum information partition (MIP) can then be defined as the partition for which normalized effective information is a minimum:The If there is more than one partition that attains the minimum normalized value, we select those partitions that generate the lowest un-normalized quantity of effective information to be the minimum information partition(s). Once the minimum information partition has been found, integrated information can be simply expressed asn binary elements φ(x1)≤n bits. This follows since the normalization is largest for the total partition, and for this partition effective information is For a discrete system composed of X, we are now in a position to identify those subsets that are capable of integrating information, the complexes. A subset S of X forms a complex when it enters state s1 if φ(s1)>0 and S is not contained in some larger set with strictly higher φ. A complex whose subsets have strictly lower φ is called a main complex. For instance, the complex in a given system with the maximum value of φ necessarily forms a main complex.For any given system In addition,φ value.At each instant in time any system of elements can be decomposed into its constituent complexes, which form its fundamental units. Indeed, only a complex can be properly considered to form a single entity. For a complex, and only for a complex, it is meaningful to say that, when it enters a particular state out of its repertoire, it generates an amount of integrated information corresponding to its X can be analyzed to find its constituent complexes, shown in shades of gray. From the figure, we see that complexes have the following properties: i) the same element can belong to more than one complex, and complexes can overlap; in particular, a smaller complex of high φ (main complex) may be contained within a larger complex of low φ; ii) a complex can be causally connected to elements that are not part of it ; iii) groups of elements with identical causal architectures can generate different amounts of integrated information depending on their ports-in and ports-out  = 0 since the interactions outside of A are redundant. Elements {n4, n5, n6} are analogous to photodiodes in a digital camera, taking a snapshot of A's state. The snapshot generates no integrated information over and above the original. Clearly an interaction occurs between elements n3 and n6, but from the perspective of the entire system it is redundant. Restricting attention to subset B, the couple, we see that integrated information generated by B is 1 bit.φ.Finding the integrated information generated by a system requires analyzing it into complexes from the ground up in terms of elementary components and elementary mechanisms or operations. n outputs. If the lower component copies the output of the upper in the previous time step then this two unit system generates n bits of integrated information – it would seem to be trivial to implement systems with arbitrarily large values of φ. However, we need to consider how such components could be built. n binary elements, and the connection between the components decomposes into couplings between pairs of elements. Analyzing the system at this more detailed level uncovers a collection of disjoint couples each of which forms an independent complex and generates 1 bit of integrated information. Since the system as a whole is disconnected, φ = 0 bits. The dotted elliptic components have been artificially imposed on the system and do not reflect the underlying causal interactions, resulting in an incorrect value of φ in the higher-level analysis. Note that, if we attempt to address this problem by adding horizontal connections between elements, so that the components are integrated within, we introduce a second problem: the horizontal couplings shrink the a posteriori repertoires of the components, reducing effective information between them. We discuss a related example, and similar considerations for continuous systems, in Consider the system in P = {X} and φ(x1) = 8 bits when the top component is firing, since it uniquely specifies the prior state of the other eight components. Increasing the number of inputs feeding into the top component while maintaining the same rule – fire if and only if all inputs are active – seems to provide a method for constructing systems with high φ using binary components. The difficulty once again lies in physically implementing a component that processes n inputs at a single point in space and at a single instant in time for large n. AND-gates. When analyzed at this level, it is apparent that the system generates 1 bit of integrated information regardless of the number of inputs that feed into the top component, since the bipartition framed by the red cut forms a bottleneck.The examples in this paper assume that the elements are abstract indivisible objects and that the rules are simple . In future work we will investigate the internal structure of elements and determine the conditions under which they can be considered to be indivisible.a posteriori repertoire of a complex X is specified using only information that is intrinsic to the complex; extrinsic inputs from the environment E = W \ X are averaged over and treated as extrinsic noise. At first glance it appears that environmental inputs cannot meaningfully contribute to the integrated information generated by X, however this is not the case.The x1. To what extent does the a posteriori repertoire of the system reflect environmental inputs?Consider the cartoon example shown in aR through cR, to the a posteriori repertoire. State a priori repertoire of states on X. Now consider the state bR simply copies aR, so aR. If aR is also copying its inputs, then the environmental inputs contribute to the information integrated by the system, albeit one temporal and spatial step removed. Row aR indirectly contributes to the total integrated information through the effect it has on bR. Finally cR specifies higher-order invariants in the a priori states of bR by combining them in some non-trivial manner. The a posteriori repertoire p(X0 → x1) reflects environmental inputs from time t = −1, and extracts higher-order invariants from environmental inputs at time t = −2. Therefore, a complex can reflect environmental inputs once they have resulted in causal interactions among its own elements.We answer the question by considering the contribution of the current state of three rows of interest, labeled bipartitions rather than all partitions of a system. It is reasonable to do so since, as shown in the We present examples and discussion, investigating the relationship between integrated information and network dynamics, causal architecture, and connectivity. Unless otherwise specified, for computational reasons we measure integrated information by considering all It can be seen from the 4 panels in the figure that inactive states – with no elements firing – are typically associated with low values of integrated information. This is because AND-gates generate less information when silent than when spiking compare .too many elements are firing. This obtains despite the fact that individual AND-gates generate more information when they are spiking than when they are silent. The reason integrated information is low when all elements are firing is that the system is “over-determined” and the whole adds nothing to the information generated by the parts. The highest values of φ occur for states with intermediate firing rates, which we refer to as balanced states . High φ means that many alternatives are ruled out by the entire system, and the parts are comparatively ineffective at specifying causes. Balanced states generate high φ because the output state of the system is highly flexible in its local causes and extremely rigid globally. The delicate trade-off between local flexibility and global rigidity justifies the term balanced.On the other hand, integrated information also decreases dramatically when the system is in a hyperactive state, with all elements firing. In a priori repertoire. In contrast, when hyperactive the systems in no integrated information. Thus, depending on the architecture of the system in question, integrated information may potentially be generated even when the system is in a fixed state and appears to be doing nothing.Note that the systems in the figure generate  integrated information even when they remain inactive for long periods. Remaining in the inactive state requires that the elements rule out alternatives from the AND-gates in the figure: they fire if a sufficiently large number of inputs are active within a given window of time. Thus, these observations may have some relevance concerning the relationship between neuronal firing rates in the thalamocortical system and consciousness. Consciousness is typically reduced when neuronal activity in the human brain is severely depressed, as under deep anesthesia or in certain comatose states. Though changes in brain function that occur in these conditions are not limited to a reduction in neuronal firing rates, the analysis of the figure indicates that integrated information would certainly suffer.To a first approximation, cortical neurons can be considered to be roughly analogous to the Consciousness also lapses when neuronal activity is excessive and hypersynchronous, as is the case in generalized seizures. The simple models shown in the figure suggest that the brain's capacity to generate integrated information would collapse also when the great majority of neurons were firing simultaneously.in vivo operate in a finely balanced regime, hovering near their spiking threshold, with excitatory and inhibitory inputs approximately canceling each other out φ. Perhaps one reason why spontaneous activity is so important is that, by ensuring the availability of a large repertoire of causal states, it represents a necessary condition for a high level of consciousness.Under normal conditions, cortical neurons φ, which reflects the bistable dynamics. φ peaks when a few of the elements are firing. However, it collapses to nearly zero when the system is hyperactive, or when the elements are shut down. The jagged form of the curve is a consequence of the network having only eight elements, resulting in abrupt dynamics.Despite its simplistic implementation, the bistable behavior of the simulated network bears some resemblance to that of thalamocortical circuits during slow wave sleep n2 has afferents from itself and n1, and fires if it receives input pattern  =  or , and is silent otherwise. Alternatively, the same dynamics can be achieved with no interactions between the elements, as in n1 alternates between firing 0 and 1. Element n2 alternates firing 00 and 11, and so forth. Element n1 has a memory of one time step, and element n4, which alternates firing eight consecutive zeros and ones, has a memory of eight time steps. The two implementations  produce identical dynamics, and cannot be distinguished by observing the systems. Nevertheless, in the first case φ = 4  or φ = .19 bits, depending on the current state, and in the second case it is zero for all states.More generally, suppose we have two systems exhibiting complex, yet identical, behavior. The first contains many interacting elements, the second is a “replay”: each element is given a list of outputs that it runs through, reproducing the original behavior. Passively observing the two systems and computing correlations, mutual information or related quantities does not distinguish between causal interactions and mere replay. In contrast, perturbing the systems and measuring integrated information does discriminate between the two situations. No integrated information is generated by the replay, and  high integrated information is generated by the causally interacting system.φ for this network we find that it is 4 bits for all states, with the total partition as the MIP. The original counting sequence was implemented in a memoryless system using the connectivity shown in n1 to n4 as the elements require more information to keep track of their position in the sequence. Nevertheless the system admits a simple description – binary counting – which the architecture reflects. The scrambled sequence does not admit a simple description and implementing it with memoryless elements requires denser wiring and more complicated Boolean functions  is shown in φ = 3.75 bits for the firing pattern shown. The network is densely and heterogeneously connected. Although every element applies the same rule, they are functionally specialized by virtue of their varying connectivity: the elements play distinct functional roles in the network, receiving unique sets of inputs, and thus specifying the a posteriori repertoire in different ways. The system is functionally integrated since it does not decompose into disconnected pieces or into weakly connected modules: the architecture tightly binds the elements, in spite of the sparse connectivity. Further, the optimized architecture is recurrent: there are multiple feedback loops embedded in the system.We have seen that, for a fixed causal architecture, the dynamics of a system determines the quantity of integrated information generated. This section shifts the emphasis and considers how integrated information depends on the underlying causal architecture. A system optimized to generate high φ for large systems, although there is evidence to suggest that architectures balancing functional specialization with functional integration produce complex dynamics φ. In the remainder of the section we investigate the general consequences of imposing structural restrictions on the class of networks under consideration (imposing strongly modular or homogeneous architectures for example) and describe the resulting information-theoretic bottlenecks, regardless of network size.It was not possible to scale the architecture up to more than a few elements because of the computational burden entailed in optimizing φ(m1) = 1.2 bits of integrated information. In contrast to the couples, the whole system does form a complex, but φ is .7 bits, even lower than for the modules. Simply connecting a collection of large integrated modules together does not ensure the resulting system will have high φ. It is necessary that the modules be properly integrated with one another. In strongly modular systems the weak or limited interactions between modules forms a bottleneck, keeping φ low as in the figure.Strongly modular systems suffer from the defect that they are insufficiently integrated. At the opposite end of the spectrum are homogeneous systems, which lack specialization. A homogeneous network has all-to-all connectivity, including self-connections. There are no limitations on the computational power of the elements, and connections efferent to different elements can have different weights, but we require that all the connections efferent to a given element be identical and that all elements implement the same computation. Under these conditions, the maximum expected integrated information generated by the system is 1 bit  proportional to n. It is known that a universal Turing machine can be built on an infinite Game of Life grid. Thus we should not be surprised that φ can increase without bound as a function of grid size: there is no limit to the computational power that can be embedded in lattice architectures. In particular, for the Game of Life, this suggests that certain configurations act in concert so the system admits a higher-order description. Forthcoming work will present a framework for analyzing systems at different spatiotemporal scales and apply it to uncover higher-order structure in the Game of Life; we will further show that the XOR-grid possesses no higher-order structure.φ using grid architectures. As a point of comparison, the 8-element AND-gate network in φ = 3.75 bits, considerably more than the maximum attained (2.3 bits) by a 3×3 grid of AND-gates. The inefficiency increases with the size of the grid; for example an XOR-grid of a million elements is needed to generate 1000 bits of integrated information. φ of a grid is limited by the interactions occurring along the perimeters of the parts, so that the expected value of φ for an n×n grid is proportional to n  is 0 or 1 according to whether the thj element fired at time t−1; and β = 1/T The temperature T is a measure of the amount of indeterminacy in the system: higher temperatures correspond to more noise. The connection matrix ijC is constructed so that the network contains certain attractors. For each attractor stored deliberately there will be additional “spurious” attractors: for example a network designed to store the firing pattern {0…01…1} will also contain its mirror image {1…10…0}. This is a quirk of the Hopfield network design. The construction is as follows. Suppose we wish to store attractor states ξ1,…Pξ. Set N dimensional space determined by the network. Choosing N to be large – hundreds of elements – and picking the attractors randomly, most easily arranges this near orthogonality. Since Hopfield networks possess all-to-all connectivity and identical elements they are similar to homogeneous systems. The crucial difference is that the weights on the arrows afferent to each element vary.A Hopfield network consists of T = .45 and initial state 11111111. The network quickly relaxes into an attractor. The graphs show integrated information as a function of temperature, which ranges between .05 and 2. We analyze integrated information generated by the system in detail for different states to better understand how integrated information and the repertoire reflect the dynamics.n1 and n8 have opposite outputs in all 6 attractors. Similarly for the pairs {n2,n7}, {n3, n6}, and {n4, n5}. It turns out that the connection matrix for the Hopfield network (embedding the 6 attractors above) has stronger connections within each couple than between them. The couples are the dominant small-scale feature of the network, a structural feature that will be reflected in the integrated information generated by the system.The specific choice of attractors has an interesting consequence. Notice that the pairs of elements φ decreases as temperature goes up. The increase in noise reduces the systems' ability to specify their a posteriori repertoires. Second, the trend across panels is that the size of the main complex decreases with increased temperature. In φ and more fundamentally in the reduction in size of the main complex.First, the graphs in φ. If we take an attractor state and alter it minimally, respecting the couple structure, we find we have to change two elements at once. There are 8 ways of doing this, producing firing patterns such as 00011110, 00101011 and so forth. These form the category of firing patterns providing the second lowest values of φ, along with a second group that we now describe. The second group contains 16 firing patterns, including 11111111; we term states in this category neutral states. The defining characteristic of neutral states is that both elements in each couple have the same output.Of all the Hopfield network's 256 possible firing patterns, integrated information is lowest when the system is in the states depicted in a priori states, and the interactions between the parts provide little extra information. Neutral states are far removed from the attractor states, and the causal architecture of the system is strongly biased against these states occurring, particularly in systems with low temperature. They are highly unstable. Nevertheless, unlike the tense states described below, φ is low. Neutral states are locally incompatible with the architecture of the system, since the elements in each couple have the same state, an outcome in opposition to the network's connectivity. The elements are working against each other locally (at the level of couples), and so the system does not cohere as a single global entity.Why do attractors and their neighbors generate low integrated information? When the system is in an attractor all of its elements are acting in concert to reinforce the current state. The parts are able to independently rule out most Tense states are the opposite of neutral states: they are locally compatible with the architecture of the system, but globally incompatible. The global tension is grounded and amplified by the local compatibility. The tense states are 01101001 and 10010110. These resemble the attractors in that each couple is in its natural, internally opposed, state. However, of all states that respect the internal structure of the couples, they are the most different from the attractor states, differing in 4 elements. Thus, the state is compatible with the couples' causal architecture, but highly incompatible with the architecture of the entire system (the relations between couples). The state depicted in φ in certain states, but φ will rapidly decrease as the system relaxes into an attractor. The system is too tightly bound to sustain interesting dynamics: the elements act in concert to push the system into its attractors. To sustain high φ it is necessary to change the connectivity so that elements act antagonistically.A massive Hopfield network with many embedded attractors may have high values of not constructed to store attractor patterns. It was optimized using a genetic algorithm, searching through networks with approximately 50% of full connectivity to find those most capable of sustaining high values of φ. The algorithm compares different networks by initializing them with a random firing pattern, running for 125 time steps and calculating φ for 10 odd firing patterns that occur. The temperature is fixed at T = .35 throughout. Over an 800 time step simulation we find that 109 of the 256 possible firing patterns arise. Of these, the 14 most common occupy slightly more than half of the running time. The system does not possess any attractors, but the dynamics are dominated by a small number of characteristic firing patterns. For each time step we compute φ, which varies with the state the system enters. The graph shows φ as a function of time; values range between .25 and 2.9 bits. Values greater than 1.1 bits occur 70% of the time, and φ of over .7 bits is generated 90% of the time. Contrast to a Hopfield network, which would remain close to an attractor over the entire run, with φ around .3 bits.The network in a posteriori repertoires. Again, although the elements all implement the same rule, the heterogeneous connectivity results in functional specialization. In addition the network is densely connected, leading to functional integration. The asymmetric, antagonistic connectivity prevents the system from relaxing into an attractor state and produces sustained “tense” dynamics in the network, and the system is thus able to consistently generate high values of integrated information. This suggests that metastable systems φ.From this example it appears that the optimization produces a similar architecture to that shown in φ, measures the information generated by a system above and beyond what can be accounted for by its parts acting independently. The subsets of a system capable of generating integrated information form complexes, which constitute individual entities and are the fundamental units into which a system decomposes. Finding the integrated information generated by a physical system requires analyzing it from the ground up, without preconception regarding the nature of its elementary units.In the paper we have extended the notions of effective information and integrated information to discrete non-stationary systems. Integrated information, φ reflects network dynamics and architecture. A few broad lessons can be extracted. First, the integrated information generated by a system depends on the current state of the system. In general, integrated information is higher when there is a balance between the number of active and inactive elements of a system. By contrast, when a system is completely inactive or hyperactive, φ values are low. Second, integrated information can differ substantially for systems with identical or similar surface dynamics, because the latter does not necessarily reflect the causal architecture of a system. For instance, a system composed of causally interacting elements can generate large amounts of integrated information, while a mere copy or “replay” of its surface dynamics generates none. More generally, integrated information appears to be a function of the complexity of the interactions leading to the observed dynamics. Third, we observed that certain classes of network architectures have low φ. Modular and homogeneous systems are unable to generate high φ because the former lack integration whereas the latter lack information. Feedforward and lattice architectures are capable of generating high φ, but they are extremely inefficient. Everything else being equal, it appears that high values of integrated information can be obtained by architectures that conjoin functional specialization with functional integration. Finally, from the probabilistic (Hopfield-style network) examples we conclude that high φ can be produced by tension between local and global interactions. Conventional Hopfield networks relax into attractor states and so cannot sustain high φ. However, random Hopfield networks can be optimized to maintain higher values of φ over the course of their dynamics.In the applications we analyzed a variety of systems to uncover how i) each conscious experience generates a huge amount of information by virtue of being one of a vast repertoire of alternatives; and ii) each conscious state is integrated, meaning that it is experienced as a whole and does not decompose into independent parts. We have shown that the way φ behaves in simple simulated networks differing in causal architecture and dynamics fits available neurobiological evidence concerning the neural substrates of consciousness. For example, φ is low for simple network analogues of inactive (“comatose”) and hyperactive (“epileptic”) states, in line with the loss of consciousness when the brain enters such states. Conversely, high φ requires balanced states similar to those observed when the brain is spontaneously active during waking consciousness. We also saw that a simplified model of bistable dynamics, loosely resembling slow-wave sleep early in the night, when consciousness fades, is not able to sustain high values of integrated information. We provided evidence that, everything else being equal, causal architectures characterized by a coexistence of functional specialization and integration are best suited to generating high values of φ, whereas strongly modular systems fare much less well. Neurobiological evidence suggests that human consciousness is generated by the thalamocortical system φ. This suggests that metastable systems, which arise when a collection of neuronal groups are loosely coupled, may be highly integrated. Intriguingly, some initial evidence obtained with multiunit recordings suggests that in awake, behaving animals populations of neurons may undergo a similar metastable dynamics The notion of integrated information is motivated by the need for a measure that captures the two basic phenomenological properties of consciousness: φ measures a process: integrated information is generated by a system transitioning from one state to the next – it does not make sense to ask about the information value of the state of a system per se. Second, φ is a causal measure: integrated information is generated only to the extent that the system transitions into a given state due to causal interactions among its elements. Thus, a system that enters a particular state due to extrinsic noise generates no integrated information, as in φ captures an intrinsic property of a system: integrated information is a function of the possible causal interactions within a system, independent of external observers. In this sense, integrated information is closer to other intrinsic properties of physical systems, such as charge or spin, than to observer-dependent properties that vary with the frame of reference, such as position or velocity. Specifically, integrated information is associated with and indeed identifies complexes – sets of elements that cannot be meaningfully decomposed into independent parts – independently of external observers. For example, elements forming two independent complexes may be lumped together into an externally defined “system” by an observer, as in A few general observations about the present measure of integrated information are also in order. First, φ for parts of the human brain, even if the connectivity and causal architecture of the neurons were known, is not a feasible undertaking, though heuristics, estimates, and relative comparisons remain possible. Applying the measure to biological system also introduces the practical issue of correctly identifying the causal architecture and a priori repertoire, a difficult empirical problem  From a theoretical perspective this problem should be addressed according to what may be called a principle of “causal ontology”: only those differences that make a difference within a system matter; differences between a priori perturbations that cannot be detected by the system can be considered as if not existing. There are a number of further issues that will be addressed in future work. A limitation of the present work is the exclusive focus on the amount of information integrated by a given network, with no consideration given to the kind of informational relationships among its elements. To address this we will move beyond quantifying integrated information as a single number and investigate the informational relationships between interacting parts by exposing the geometry of the causal interactions. Another shortcoming is that it focuses exclusively on memoryless systems, in which integrated information can only be generated over one time step. In a forthcoming paper we will coarse-grain discrete systems and develop techniques to find the natural spatiotemporal scale at which a system generates integrated information. This will allow us to deal with systems with memory, as well as to make a first step towards analyzing large-scale, hierarchically organized systems. Finally, the networks considered here were analyzed as isolated entities, without consideration for their environment (or rather by averaging over possible extrinsic inputs). In future work we will discuss how discrete systems interact with and incorporate information from the environment, as well as the relationship between integrated information and learning.Finally, we should mention some of the many limitations of the present work. Foremost among them is that our examples are restricted to small-scale models, so it is unclear to what extent the principles suggested by our partial explorations would scale with larger networks. The impossibility of measuring integrated information for larger systems is due to the combinatorial explosion in the partitions of a system as the number of elements is increased. An inevitable consequence is that computing Text S1Supplementary Information(0.55 MB DOC)Click here for additional data file.