Intra-subject Elastic Registration of 3D Ultrasound Images 3D registration of ultrasound images is an important and fast-growing research area with various medical applications, such as image-guided radiotherapy and surgery. However, this registration process remains extremely challenging due to the deformation of soft tissue and the existence of speckles in these images. This paper presents a technique for intra-subject, intra-modality elastic registration of 3D ultrasound images. Using the general concept of attribute vectors, we define the corresponding voxels in the fixed and moving images. Our method does not require presegmentation and does not employ any numerical optimization procedure. As the computational requirements are minimal, the method has potential use in real-time applications. The technique is implemented and tested on 3D ultrasound images of human liver, captured by a 3D ultrasound transducer. The results show that the method is sufficiently accurate and robust even in cases where artifacts such as shadows exist in the ultrasound data. Ultrasound is an easy-to-use, inexpensive and real-time imaging tool, with no ionizing radiation, which is widely utilized in diagnosis applications. It is often employed to scan soft tissues such as the liver, kidneys and pancreas.. The soft tissues are easily deformed during scanning, mainly because of transducer pressure, respiration, and heartbeat. Determination of this nonrigid deformation is crucial for real-time applications such as image-guided surgery, radiotherapy, and robot-assisted biopsy. Intra-operative registration may be employed to compensate for patient's breathing, or to track a tumor. Furthermore, ultrasound registration could be used to measure tissue mechanic properties. However, due to speckles, shadowing effects and the poor quality of the images, the registration of this imaging modality is a challenging process especially for large nonrigid deformations. Consequently, the research on intramodality registration of ultrasound images is very limited in the literature, but there appears to be an accrued interest in work related to the application of ultrasound images (Pluim and Fitzpatrick (2003) ). Previous attempts include the development of techniques for the registration of ultrasound to other image modalities such as CT and MRI (Blackall et al. (2005) ; Letteboer et al. (2005) ; Aylward et al. (2002) ; King et al. (2001) ; Arbel et al. (2001) ; King et al. (2000) ; Pagoulatos et al. (1999) ; Maintz and Viergever (1998) ), as well as intra-modality registration of ultrasound images discussed in the following. General information on registration of other modalities can be found in Hill et al. (2001) ; Hajnal et al. (2001) ; Maintz and Viergever (1998) .. As the deformation of soft tissue is not guaranteed to occur in the plane scanned by the ultrasound beam, 2D registration of ultrasound images is not sufficiently robust for most clinical applications. The anatomical features might appear or disappear in the 2D images as a result of out-of-plane movement. Thanks to techniques such as free-hand 3D ultrasound reconstruction and newly developed 3D ultrasound transducers, there has been growing interest in the registration of 3D ultrasound images. The following introduces some of the prior art on intra-modality registration of ultrasound images especially the 3D nonrigid techniques. The type of registration (Rigid, Affine, Elastic, etc.) , the targeted application, the validation experiments and the results of each method are briefly discussed. Rohling et al. (1998) rigidly registered the reconstructed 3D ultrasound images of human gall bladder to exploit spatial compounding in order to improve the quality of 3D ultrasound data. The correlation of the gradient magnitude data of the two volumes is maximized to find the registration parameters. It is shown that spatially compounded datasets yield better segmentation results. The algorithm was limited only to applications where rigid-body registration would provide acceptable accuracy and where computational time in the order of hours was reported. Shekhar and Zagrodsky (2002) investigated the registration of ultrasound volumes, based on the mutual information measure. Median filter was found to be the proper smoothing filter for this method. Intensity quantization was shown to increase the signal-to-noise ratio, and robust optimization behavior was achieved using trilinear partial volume distribution interpolation. The method was applied to rigid-body transformation with both uniform and nonuniform scaling as well as to affine transformation. To evaluate the method, the deformations were applied to five volume pairs of cardiac ultrasound data. The results, only reported for simulated deformations applied to the ultrasound data, demonstrate a large capture range, high accuracy, and execution times in the order of 16-24 min for rigid body registration on volumes with the size of 128×128×512 voxels and in the order of minutes for volumes with the size of 64×64×256 voxels (note: to fairly compare computation time mentioned in each referenced article, one has to consider the platform algorithm has been implemented on).. In another study, Krucker et al. (2002) developed a subvolume-based algorithm for elastic ultrasound registration. The image was divided into subvolumes interactively, and local rigid registrations were computed. Connectivity of the entire volume was ensured by global interpolation using thin-plate splines after each iteration. At the first stage, a global affine registration was optionally determined using MIAMI Fuse c software (Meyer et al. (1997) ). The performance was tested on synthetic deformations of 1.5-2.5 mm and two phantom scans, simulating the deformations in breast images. In this approach, the average deformation of 1.01 mm between the two volumes was reduced to 0.30 mm. The algorithm computation time was reported to be below 5 min for ultrasound volumes as large as 13 MB. No result was reported for larger initial average displacements. The subvolume-based approach was also employed for spatial compounding and extended field-of-view (Poon and Rohling (2006) ; Groves and Rohling (2004) ).. Investigating a correlation-based approach, Xiao et al. (2002) registered 3D free-hand ultrasound images of breast using a block-matching scheme. The reference image was sampled at a set of grid points. Each sample comprised a block of voxels centered on a grid point. The best match for the block was subsequently located within a 3D search window. The compounding results of two sweeps before and after registration were reported. The algorithm was tested on real breast cyst images, with initial deformations of only a few millimeters, and computational time in the order of minutes was reported.. Deformable registration of cardiac images has been investigated more specifically in Ledesma-Carbayo et al. (2001) ; Shekhar et al. (2004) ; Makela et al. (2002) . Considering elastic registration as an optimization problem, LedesmaCarbayo et al. (2001) elastically registered the 2D sequences of cardiac images by optimizing a sum of squared differences with respect to the coefficients of splines. Shekhar et al. (2004) is a continuation of work reported in Shekhar and Zagrodsky (2002) , however no specific information was provided on the computational complexity of the algorithm. 3D-ultrasound registration has also been targeted for brain shift correction. In Pratikakis et al. (2001) , reg-istration was defined as a motion estimation problem. A cost function was minimized in a multi-scale framework. A coarse-to-fine registration approach was proposed and tested on only one 3D ultrasound volume of an 8-month old baby. No detailed analysis was forthcoming as to the execution time of the technique.. In this paper, we present a new intra-subject, intra-modality elastic registration technique of 3D ultrasound images. The algorithm is designed to be automatic, fast and computationally efficient, therefore potentially applicable for real-time image-guided surgery and radiotherapy. The current paper builds upon our earlier report (Foroughi and Abolmaesumi (2005) ) and provides a rigorous examination and analysis of the accuracy and performance of the algorithm implemented on real 3D ultrasound datasets. Under our approach, the concept of attribute vectors is employed to find corresponding voxels in the two ultrasound volumes being registered. Using the attribute vectors, a set of leading points is automatically selected in the deformed image (referred to as moving image), and the corresponding points are found in the reference or base image (referred to as fixed image). Finally, the moving image is warped into the fixed image. The registration results on human liver images show that the algorithm is accurate for initial deformations in the order of one centimeter with computational time of 3 minutes or less on a conventional PC.. Our approach although similar to that of HAMMER (Shen and Davatzikos (2002); Shen (2004) ), which was originally proposed for the registration of MRI images of human brain, specifically differs in the following: 1) The computational time of the HAMMER technique has been reported to be in the order of hours, which is not acceptable for real-time clinical applications. The high computational cost is attributable to the calculation of Geometric Moment Invariant (GMI) attribute vector elements and to the employment of an iterative optimization in the registration process. To reduce the computation time, we propose a new set of elements for the attribute vector and eliminate the numerical optimization phase in our implementation. We demonstrate that our new attribute vector produces superior registration results in the case of ultrasound images, compared to the results achieved by using GMIs. Furthermore, the registration accuracy is sufficient for the majority of clinical applications, without the employment of the iterative phase; 2) Our energy function is simpler than the one used in HAMMER, and this also reduces our computational complexity; and 3) A different warping method is employed to consider a more nonuniform distribution of leading points in the ultrasound images of human liver in comparison to the MRI images of brain. This paper is organized as follows. Section 2 outlines the methodology and details of the implementation. Experimental results on 3D liver images are presented in Section 3. Section 4 discusses the sensitivity of the algorithm performance to the effect of parameters, and provides a comparison with the GMI-based feature vectors. Section 5 concludes the paper and presents the future work. We have developed a method for 3D elastic registration of ultrasound images, which is intended eventually for radiotherapy and surgical applications. The algorithm does not require presegmentation or numerical optimization resulting in relatively shorter running time. A number of experiments were carried out using naturally and artificially deformed ultrasound images of liver showing clinically acceptable accuracy. Detailed analyses were also provided on the selection of parameters for the algorithm and on the discrimination ability of the proposed attribute vector. Future work will include improving performance by developing a method for failure detection of registration of leading points, as well as improving the normalization strategies. In addition, the code could be optimized to reduce the running time of the algorithm to within an acceptable range for intra-operative applications. Real-time implementation also seems to be achievable by exploiting hardware acceleration. Finally, the algorithm should be tested for the registration of ultrasound images of other anatomical parts in addition to the liver. For a successful registration, the correct values for the parameters of the algorithm should be selected. This section investigates the effect of these parameters on registration results and the sensitivity of the registration to the parameters. Moreover, additional experiments are provided to show the discrimination ability of the proposed attribute vector and to compare the performance of the proposed vector elements with the GMIs for ultrasound images. Table 1 The simulation experiment was carried out for a 185x113x199 ultrasound volume of a liver using a Pentium 4 2.8GHz CPU. average initial displacement of the deformation matrix was 13.02 voxels or 9.24 mm, and the maximum displacement was 27.71 voxels. The The algorithm has been implemented in MATLAB TM . The matching and warping parts are written in C, and compiled with the C compiler of MATLAB TM to reduce the computation time. Two sets of tests were carried out in order to evaluate the accuracy, robustness, and speed of the technique. First, actual 3D ultrasound images were captured, and registered to each other. Registration of naturally deformed images is the best method for testing the performance of the registration algorithm since some issues, such as motion ambiguities and speckle decorrelation (Pratikakis et al. (2001) ), are not present in simulation experiments. However, simulation experiments help to evaluate the accuracy of registration since exact deformations are unknown for actual images. Hence, in the second set of tests, the ultrasound volumes were elastically deformed by a known displacement matrix. The original and deformed volumes were then registered to each other. The proposed algorithm is demonstrated in Figure 1 . Initially, both fixed and moving images are passed through a Gaussian filter with the size of 7×7×7 voxels. This filter is an anti-aliasing filter for down-sampled images which reduces the effect of speckles. In the next step, the elements of attribute vectors are calculated in two different scales for each voxel in both fixed and moving images. Then, the leading points with the highest magnitude attribute vectors are selected and registered, and finally the moving image is warped to the fixed image. The following paragraphs explain these steps in detail.

O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Smallest Number of Sensors for k-Covering This paper presents some theoretical results on the smaller number N k (a, b) of sensors to achieve k coverage for the rectangular area Wireless Sensor Networks (WSN) consist of a large number numbers of sensors distributed uniformly in a target area, which monitor in a cooperative manner the physical word. Sensors are in fact small devices capable of sensing variations in temperature, light, gas, motion etc, computing and storing information and communicating with the neighbour sensors. The technology has nowadays made it possible to produce these devices at a cheap price so that WSN networks are now involved in various applications from agriculture, surveillance, asset tracking, health care to building safety and evacuation.. An important aspect in WSN applications is the coverage problem, which investigates how well the target area is monitored by sensors. If each point of the area is covered by at least k sensors then the WSN network is said to be k-covered, where k is the degree of the coverage. For example, tracking WSN networks are at least 3-covered as they use triangulation. Moreover, most WSN networks must be at least 2-covered to assure the robustness property.. It is clear that the bigger the coverage degree is, the more sensors must be used in WSN networks. So far, all research works on the WSN coverage problem have assumed that a large number of sensors is distributed in the target area to assure the network is k-covered. However, no information has been given about the number of sensors to use for the target area or equivalently about the sensor density. This article comes to investigate some properties of the minimal number of sensors and to provide un upper bound for the minimal density of sensors to achieve k-coverage in a rectangular area. Investigating 1-coverage or circle packing has been researched as geometrical combinatorial since 1939. Researchers have tried to find mathematical equations for the optimal 1-covering or even to prove that some configurations are optimal. For example Kershner [4] investigated the problem of covering any 2D set of points with similar circles based on some geometrical combinatorial techniques. This early work proved that the minimal number of circles N (ε) of radius ε to cover a close set of point M satisfies. where |M | denotes the area of closed by M . The result was proven by using a double inequality for the quantity πε 2 N (ε) representing the total area covered by the circles. An important consequence of this result is that the proportion of unavoidable overlapping can be approximated by 2π √ 3 9 ≃ 1.209. We can also mention the early work of Verblunsky [10] who proved that the minimum number N (l) of circles of radius 1 to cover a square of length l should satisfy. These two results come to suggest that the sensor density for 1-coverage can be estimated by. However, these early works [4] , [10] do not provide any information about the pattern of circles used to achieve the minimal coverage. Recently, several articles on circle packing problems investigated efficient ways to cover a rectangle with similar circles (see [6] , [8] or [9] amongst others). These geometrical combinatorics researches confirmed that optimal packing is difficult to be achieve even for small number of circles. Furthermore, no pattern was detected for the packing configuration that gives optimality.. Recently, several papers have dealt with the k-coverage problems in the context of sensor networks studying conditions when this is achieved or algorithms to detected when this happens. Some of these contributions have made marginal reference to the minimum number or equivalently to the minimum density of sensors that assures k-covering of a given area. Generally, all these works have considered that the number of sensors to use is big enough to k-cover the target area. Adlakha and Srivasyava [1] developed an exposure-based model to find the sensor density required to achieve full coverage of a given area. They proved that the number of sensors to achieve 1-covering is in the order of O(A/r 2 ), where A is the area to cover however they did not provide any constant for the magnitude of A/r 2 . Ammari and Das [2] investigated the problem of k-coverage proposing a condition to achieve it. They considered the target area divided in "Reuleaux" triangles which are formed by the intersection of 3 circles. The main result of their work states that the target area is k-covered if and only if each "Reuleaux" triangle contains at least k-active sensors. Another important results proposed by Ammari and Das gives that the minimal density of sensors to guarantee k-coverage is λ(r, k) =. , where r is the radius of the sensing disks.. Most covering problems present huge difficulties to solve or to derive a polynomial algorithm even in particular cases like regular or simple shapes and lower dimensional space. The 2D problem of covering a bounded domain with arbitrary shaped objects was proven to be exponential on the size of the packing space [7] . The particular case of covering any polygon with n similar disks is known to be NP-hard [5] . Consequently, the problem of finding the least number of disks to k-cover a rectangle is NP-hard.. All these works have shown that calculating the minimal number of circles to pack a rectangle is a hard problem and there is no patter associated with this covering. Moreover, the results concerning the minimal number of circles for k-coverage are all either asymptotic based on some limits or approximative based on some inequalities.

O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


RONoC: A Reconfigurable Architecture for Application-Specific Optical Network-on-Chip Optical Network-on-Chip (ONoC) is a promising emerging technology, which can solve the bottlenecks faced by electrical on-chip interconnection. However, the existing proposals of ONoC are mostly built on fixed topologies, which are not flexible enough to support various applications. To make full use of the limited resource and provide a more efficient approach for resource allocation, RONoC (Reconfigurable Optical Network-on-Chip) is proposed in this letter. The topology can be reconfigured to meet the requirement of different applications. An 8×8 nonblocking router is also designed, together with the communication mechanism. The simulation results show that the saturation load of RONoC is 2 times better than mesh, and the energy consumption is 25% lower than mesh. Network-on-Chip is a promising way to integrate multi-core system on a single chip [1] , [2] . Compared with the conventional electronic interconnect, optical interconnect is capable of providing more bandwidth at a lower power budget. Therefore, photonic technology is becoming an increasingly attractive solution to the various challenges faced by on chip communication. Meanwhile, recent developments in silicon photonics have made it feasible to fabricate all the necessary optical building blocks on a single chip [3] , achieving high-bandwidth and energy-efficient intra-chip communications. Various architectures have been proposed, most of which are established on fixed topology [4] - [7] . However, the NoC architecture may face various applications which own different traffic characteristics. Thus, the ONoC proposed in literature often lacks flexibility to meet the different requirements of the applications.. On the other hand, since the technology of optical buffers is not so mature in today's photonic technology, optical packet switching is difficult to implement for on chip networks. Hence, a circuit-switching-like transmission mechanism is often employed for ONoC. However, limited path diversity is one of the main disadvantages, especially for the mesh based ONoC. The high contention probability leads to high latency and degraded throughput. One solution is to employ the concept of reconfiguration to meet different requirement of various applications [8] , [9] . It can also help [11] . In the optical domain, related researches are relatively few. Artundo et al.. propose a reconfigurable optical interconnect and its topology in optical layer is adapted automatically to the evolving traffic situation, allowing a large fraction of the short coherence messages to use the optical links [12] . Gao et al. present the RePNoC architecture which is optional to preset the topology by configuring the connections according to different application patterns [13] . The connections between source and destination IP cores with long links can be set up by bypassing the intermediate routers.. In this letter, a reconfigurable architecture for application-specific optical Network-on-Chip is proposed, the interconnection of which can be reconfigured for different applications, achieving lower latency and power consumption. RONoC topology has a simple architecture and is easy to configure the connections just by altering the states of different groups of microring resonators (MRs). A configuration algorithm for different traffic patterns is also developed, which applies universally to diverse applications. The simulation results prove that RONoC provides a lowlatency and energy-efficient performance. The RONoC architecture is proposed to enable the topology to dynamically match the communication patterns of various applications. The reconfiguration operation is simple, which just needs to power on or off the related resonators. A configuration algorithm for RONoC is developed to achieve lower power consumption. The simulation results show that RONoC has a much higher throughput and smaller power consumption. In our future work, some dynamic traffic patterns, like all to all communication, will be considered. New configuration algorithm will be developed.

O O O O O B B I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Extraction and Recognition Methods of Vehicle Driving Tendency Feature Based on Driver-vehicle-environment Dynamic Data under Car Following Driving tendency is the comprehensive embodiment of driver's physiological and psychological characteristics. It reflects driver's psychological condition during the process of driving. Identification of the driving tendency has great significance to researching of the driver assistant system, especially the active security warning system. In this paper, data from simulative driving and real driving experiment of different drivers (risk-taking, cautious and conservative type, etc) under the state of car-following flow are obtained through questionnaires on psychological test. The feature eigenvectors with good ability to classify the type of driving tendency is extracted based on rough set theory. Dynamic recognition method of driving tendency is put forward with model calibration and verification making use of real data. The results of verification are compared with the results of the questionnaires. It is shown that the pattern recognition model is feasible, and can realize real-time recognition of the driver tendency under the state of car-following. The driver's internal factors are important to impacting traffic safety in addition to vehicle-road-environment. The driver's internal factors include the driver's physiological and psychological characteristic. The physiological characteristic is the basis of the psychological characteristic and is reflected in the psychological characteristics. The differences of drivers' age, gender, driving experience and individuality result in the different psychological characteristics. And the different psychological characteristics are reflected as driving tendencies. The research of driving tendency recognition plays an important role in improving the applicability and accuracy of vehicle active safety systems (collision avoidance warning system, for example). Former researches focus mainly on driver characteristics and its impact on driving behavior [1] [2] [3] [4] [5] , deducing and forecasting of driving behavior from human, vehicle and environment factors [6] [7] [8] [9] . At present, the researches of driving tendency focus mainly on driver psychological measurement and analysis of the influence on macroscopic traffic safety in a relatively static condition [10] [11] . Some researchers has introduced driver individual differences to vehicle active safety system. But there is still no detailed study on how to distinguish the driver's individual differences.. The driver's cognition and processing of the environment information (That is, Driver Behavior Characteristics) are influenced by physiological and psychological characteristics which are called driving tendency. And physiological and psychological characteristics can be gotten from driver's behavior [12] [13] [14] . Driving tendency recognition in driving-assistance system is within the domain of artificial psychology and emotion recognition [15] [16] . So we can extract characteristic variables of driving tendency, and identify its type. The data of various driving tendency types (such as risk-taking, cautious and conservative)are obtained in the car-driving state through psychological questionnaire tests, observed experiments and real vehicle experiments. The feature extraction method based on rough set theory is used to extract eigenvectors with good ability to classify the type of driving tendency. The determinant of driving tendency is gotten through vehicle kinematics theory. The dynamic recognition method of driving tendency is put forward with model calibration and verification made use of real driving data. For simplicity, considering the intrusion of contact inspection to drivers, this paper focuses mainly on the driving tendency identification based on coordinate deduction of vehicle and road. Driving tendency is the comprehensive embodiment of driver's physiological and psychological characteristics. It reflects driver's psychological condition during the process of driving. In this paper, the feature extraction method based on rough set theory is used to extract eigenvectors with good ability to classify the type of driving tendency. Dynamic recognition model of driving tendency is put forward with calibration and verification making use of real data. Results show that the recognition method is feasible, and can realize realtime recognition of the driver tendency under the state of car-following. It can provide theoretical basis for the realization of automobile active safety system personalized.

O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A high throughput robot system for machine vision based plant phenotype studies This work demonstrates how a high throughput robotic machine vision systems can quantify seedling development with high spatial and temporal resolution.The throughput that the system provides is high enough to match the needs of functional genomics research. Analyzing images of plant seedlings growing and responding to stimuli is a proven approach to finding the effects of an affected gene. However, with 10 4 genes in a typical plant genome, comprehensive studies will require high throughput methodologies. To increase throughput without sacrificing spatial or temporal resolution, a 3 axis robotic gantry system utilizing visual servoing was developed. The gantry consists of direct drive linear servo motors that can move the cameras at a speed of 1 m/s with an accuracy of 1 µm, and a repeatability of 0.1 µm. Perpendicular to the optical axis of the cameras was a 1 m 2 sample fixture holds 36 Petri plates in which 144 Arabidopsis thaliana seedlings (4 per Petri plate) grew vertically along the surface of an agar gel. A probabilistic image analysis algorithm was used to locate the root of seedlings and a normalized gray scale variance measure was used to achieve focus by servoing along the optical axis. Rotation of the sample holder induced a gravitropic bending response in the roots, which are approximately 45 µm wide and several millimeter in length. The custom hardware and software described here accurately quantified the gravitropic responses of the seedlings in parallel at approximately 3 min intervals over an 8-h period. Here we present an overview of our system and describe some of the necessary capabilities and challenges to automating plant phenotype studies. A standard approach to determining the function of a gene is to discover the consequences of its mutation, which constitute the phenotype. For the reference species Arabidopsis thaliana, almost all of its approximately 25,000 genes have been mutated and seeds of most mutant plants are publicly available. However, phenotypes have been detected and described for perhaps as few as 10 % of those genes. Even in those relatively few cases, better quantification of the phenotypes and discovery of additional phenotypes in the mutants would improve understanding of the affected genes. Adding the time domain to the analysis, allowing the development of a phenotype to be captured has proven very useful in determining details of gene function. Machine vision technology promises to be effective in the effort to improve plant phenotype research [7, 10, 11, 15, 16, 24] , where the experiments call for micron level measurements of seedlings on time scales of minutes or daily measurements of large crop plants in a greenhouse.. To be effective at the genome scale, the measurement platform must have the capacity to characterize thousands of plants in weeks or months. This need for throughput cannot be at the expense of spatial or temporal resolution. Parallelizing data acquisition by replicating the imaging system can increase throughput while keeping resolution and sample frequency constant, but the need to frame and focus each subject manually at the start of an experiment would be a daunting and highly repetitive task if carried out for a large number of imaging systems. This limits the scalability and increases the cost of this solution. As the images are acquired only at certain time points the imaging systems are idle for most of the duration of data gathering.. The repetitive tasks associated with collecting time series of consistently framed and focused images of multiple plants are well suited for automation by robotics. One robotic approach is to move the plants repeatedly through a fixedcamera imaging station [14] . However, moving the plants can potentially alter their growth and behavior at the fine scales intended to be captured and analyzed. This drawback can be avoided by moving the imaging device across a fixed grid of samples. For the moving camera solution to be effective, several machine vision application challenges must be solved. These include rapid and accurate visual servoing, object recognition, and real-time image-based focusing. This report describes the construction and performance of a gantry robot system (Fig. 1 ) that meets these challenges. The application presented is the tracking of root growth in seedlings of Arabidopsis thaliana (Fig. 2 ) after reorientation by 90 • to induce a curvature response known as gravitropism. The measurement throughput, spatio-temporal resolution, precision, and amortized cost of the robot system described here provides a strong rationale for adopting robotic platforms for plant phenotyping studies.. The remainder of this paper is organized as follows, Sect. 2 describes the hardware of the robotic system and the sample holding fixture. A brief description of a typical data gathering session is provided in Sect. 3 . Section 4 provides details of the image processing used to locate the seedling root and root tip. Section 5 assesses the performance of the platform for monitoring a typical root gravitropic experiment. Section 6 provides an outline of future work.

O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Space Complexity vs. Query Complexity Abstract. Combinatorial property testing deals with the following relaxation of decision problems: Given a fixed property and an input x, one wants to decide whether x satisfies the property or is "far" from satisfying it. The main focus of property testing is in identifying large families of properties that can be tested with a certain number of queries to the input. Unfortunately, there are nearly no general results connecting standard complexity measures of languages with the hardness of testing them. In this paper we study the relation between the space complexity of a language and its query complexity. Our main result is that for any space complexity s(n) ≤ log n there is a language with space complexity O(s(n)) and query complexity 2 Ω(s(n)) . We conjecture that this exponential lower bound is best possible, namely that the query complexity of a languages is at most exponential in its space complexity. Our result has implications with respect to testing languages accepted by certain restricted machines. Alon et al. [FOCS 1999] have shown that any regular language is testable with a constant number of queries. It is well known that any language in space o(log log n) is regular, thus implying that such languages can be so tested. It was previously known that there are languages in space O(log n) which are not testable with a constant number of queries and Newman [FOCS 2000] raised the question of closing the exponential gap between these two results. A special case of our main result resolves this problem as it implies that there is a language in space O(log log n) that is not testable with a constant number of queries, thus showing that the o(log log n) bound is best possible. It was also previously known that the class of testable properties cannot be extended to all context-free languages. We further show that one cannot even extend the family of testable languages to the class of languages accepted by single counter machines which is perhaps the weakest (uniform) computational model that is strictly stronger than finite automata. Basic Definitions: Combinatorial property testing deals with the following relaxation of decision problems: given a fixed property P and an input x, one wants to decide whether x satisfies P or is "far" from satisfying the property. This notion was first introduced in the work of Blum, Luby and Rubinfeld [5] , and was explicitly formulated for the first time by Rubinfeld and Sudan [16] . Goldreich, Goldwasser and Ron [8] have started a rigorous study of what later became known as "combinatorial property testing". Since then much work has been done, both on designing efficient algorithms for specific properties, and on identifying natural classes of properties that are efficiently testable. For detailed surveys on the subject see [6, 15] .. In this paper we focus on testing properties of strings, or equivalently languages 4 . In this case a string of length n is -far from satisfying a property P if at least n of the string's entries should be modified in order to get a string satisfying P. An -tester for P is a randomized algorithm that given and the ability to query the entries of an input string, can distinguish with high probability (say 2/3) between strings satisfying P and those that are -far from satisfying it. The query complexity q( , n) is the maximum number of queries the algorithm makes on any input of length n. Property P is said to be testable with a constant number of queries if q( , n) can be bounded from above by a function of only. For the sake of brevity, we will sometimes say that a language is easily testable if it can be tested with a constant number of queries 5 . If a tester accepts with probability 1 inputs satisfying P then it is said to have a 1-sided error. If it may err in both directions then it is said to have 2-sided error. A tester may be adaptive, in the sense that its queries may depend on the answers to previous queries, or non-adaptive, in the sense that it first makes all the queries, and then proceeds to compute using the answers to these queries. All the lower bounds we prove in this paper hold for the most general testers, namely, 2-sided error adaptive testers.. Background: One of the most important questions in the field of property testing is to prove general testability results, and more ambitiously to classify the languages that are testable with a certain number of queries. While in the case of (dense) graph properties, many general results are known (see [2] and [3] ) there are not too many general results for testing languages that can be decided in certain computational models. Our investigation is more related to the connection between certain classical complexity measures of languages and the hardness of testing them, which is measured by their query complexity as defined above. A notable result in this direction was obtained by Alon et al. [1] were it was shown that any regular language is easily testable. In fact, it was shown in [1] that any regular language can be tested with an optimal constant number of queries Θ(1/ ) (the hidden constant depends on the language). It has been long known (see exercise 2.8.12 in [14] ) that any language that can be recognized in space 6 o(log log n) is in fact regular. By the result of [1] this means that any such language is easily testable. A natural question is whether it is possible to extend the family of easily testable languages beyond those with space complexity o(log log n). It was (implicitly) proved in [1] that there are properties in space O(log n) that are not easily testable, and Newman [13] raised the question of closing the exponential gap between the o(log log n) positive result and the Ω(log n) negative result. Another natural question is whether the family of easily testable languages can be extended beyond those of regular languages by considering stronger machines. Newman [13] has considered nonuniform extensions of regular languages and showed that any language that can be accepted by read-once branching programs of constant width is easily testable. Fischer and Newman [7] showed that this can not be further extended even to read twice branching programs of constant width. For the case of uniform extensions, it has been proved in [1] that there are context-free languages that are not easily testable.. In this paper we study a relation between the space complexity and the query complexity of languages. As a special case of this relation we resolve the open problem of Newman [13] concerning the space complexity of the easily testable languages. We also show that the family of easily testable languages cannot be extended to essentially no family of languages accepted by (uniform) machines stronger than finite state automata.. Main Results: As we have discussed above there are very few known connections between standard complexity measures and query complexity. Our first and main investigation in this paper is about the relation between the space complexity of a language and the query complexity of testing it. Our main result shows that in some cases the relation between space complexity and query complexity may be at least exponential. As we show in Theorem 2 below, it can be shown that there are languages, whose space complexity is O(log n) and whose query complexity is Ω(n). Also, as we have previously noted, languages whose space complexity is o(log log n) can be tested with Θ(1/ ) queries. Therefore, the interesting space complexities s(n) that are left to deal with are in the "interval" [Ω(log log n), O(log n)]. For ease of presentation it will be easier to assume that s(n) = f (log log n) for some integer function x ≤ f (x) ≤ 2. x . As in many cases, we would like to rule out very "strange" complexity functions s(n). We will thus say that s(n) = f (log log n) is space constructible if the function f is space constructible, that is, if given the unary representation of a number x it is possible to generate the binary representation of f (x) using space O(f (x)). 6 Throughout this paper we consider only deterministic space complexity. Our model for measuring the space complexity of the algorithm is the standard Turing Machine model, where there is a read only input tape, and a work tape where the machine can write. We only count the space used by the work tape. See [14] for the precise definitions. For concreteness we only consider the alphabet {0, 1}.. Note that natural functions, such as s(n) = log 2 log n and s(n) = √ log n are space constructible 7 .. Theorem 1 (Main Result). For any (space constructible) function s(n) there is a language in space O(s(n)), whose query complexity is 2 Ω(s(n)) .. We believe it will be interesting to further study the relation between these two measures. Specifically, we raise the following conjecture claiming that the lower bound of Theorem 1 is best possible:. Conjecture 1. Any language in space s(n) can be tested with query complexity 2 O(s(n)) .. As we have mentioned above, as one of the steps of the proof of Theorem 1 we prove the following result that may be of independent interest.. There is a language in space O(log n), whose query complexity is Ω(n).. To the best of our knowledge, the lowest complexity class that was previous known to contain a language, whose query complexity is Ω(n), is P (see [9] ). If Conjecture 1 is indeed true then Theorem 2 is essentially best possible.. As an immediate application of Theorem 1 we deduce the following corollary, showing that the class of easily testable languages cannot be extended from the family of regular languages even to the family of languages with space complexity O(log log n) thus answering the problem raised by Newman in [13] concerning the space complexity of easily testable languages. Corollary 1. For any k > 0, there is a language in space O(log log n), whose query complexity is Ω(log k n).. Corollary 1 rules out the possibility of extending the family of easily testable languages from regular languages, to the entire family of languages, whose space complexity is O(log log n). We turn to address another result, ruling out another possible extension of regular languages. As we have mentioned before, it has been shown in [1] that there are context-free languages that are not testable. Hence, a natural question is whether there exists a uniform computational model stronger than finite state machines and weaker than stack machines such that all the languages that are accepted by machines in this model are testable. Perhaps the weakest uniform model within the class of context-free languages is that of a deterministic single-counter automaton (also known as one-symbol push-down automaton). A deterministic single-counter automaton is a finite state automaton equipped with a counter. The possible counter operations are increment, decrement and do nothing, and the only feedback from the counter is whether it is currently 0 or positive (larger than 0). Thus, such an automaton, running on a string ω reads an 7 Note that when s(n) = log 2 log n we have f (x) = x 2 and when s(n) = √ log n we have f (x) = 2 x/2 . We also note that we use the standard notion of space constructibility, see e.g. [14] .. input character at a time, and based on its current state and whether the counter is 0, jumps to the next state and increments/decrements the counter or leaves it unchanged. Such an automaton accepts a string ω if starting with a counter holding the value 0 it reads all the input characters and ends with the counter holding the value 0. It is quite obvious that such an automaton is equivalent to a deterministic push-down automaton with one symbol stack (and a readonly bottom symbol to indicate empty stack). This model of computation can recognize a very restricted subset of context free languages. Still, some interesting languages are recognized by such an automaton, e.g. D 1 the first Dyck language, which is the language of balanced parentheses. Formal definition and discussion on variants of counter automata can be found in [18] .. In this paper we also prove the following theorem showing that the family of testable properties cannot be extended even to those accepted by single-counter automata.. Theorem 3. There is a language that can be accepted by a single-counter automaton and whose query complexity is Ω(log log n).. Combining Theorem 3 and Corollary 1 we see that the family of testable properties cannot be extended beyond that of the regular languages in two natural senses.. Organization: The rest of the paper is organized as follows. In Section 2 we prove the exponential relation between space complexity and query complexity of Theorem 1. An important step in the proof is Theorem 2 that we also prove in this section. Section 3 contains the proof of Theorem 3 showing that there are languages accepted by counter machines that are not easily testable. Section 4 contains some concluding remarks and open problems. Due to space limitations several proofs are omitted and will appear in the full version.

O B O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O B O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Relating reflex gain modulation in posture control to underlying neural network properties using a neuromusculoskeletal model Abstract During posture control, reflexive feedback allows humans to efficiently compensate for unpredictable mechanical disturbances. Although reflexes are involuntary, humans can adapt their reflexive settings to the characteristics of the disturbances. Reflex modulation is commonly studied by determining reflex gains: a set of parameters that quantify the contributions of Ia, Ib and II afferents to mechanical joint behavior. Many mechanisms, like presynaptic inhibition and fusimotor drive, can account for reflex gain modulations. The goal of this study was to investigate the effects of underlying neural and sensory mechanisms on mechanical joint behavior. A neuromusculoskeletal model was built, in which a pair of muscles actuated a limb, while being controlled by a model of 2,298 spiking neurons in six pairs of spinal populations. Identical to experiments, the endpoint of the limb was disturbed with force perturbations. System identification was used to quantify the control behavior with reflex gains. A sensitivity analysis was then performed on the neuromusculoskeletal model, determining the influence of During posture control, humans have two strategies to counteract unexpected disturbances and keep their equilibrium position. Co-contraction of the muscles increases joint viscoelasticity, while maintaining equilibrium. Co-contraction provides instantaneous viscoelasticity at the expense of high metabolic energy consumption. Sensory feedback is energy efficient, although effectiveness is limited due to the inherent neural time delays.. In the presence of fast unpredictable disturbances in the upper extremity, reflexes through afferent feedback provided by muscle spindles and Golgi tendon organs are the major contributors to sensory feedback. Muscle spindles are located in the muscles and provide feedback on stretch and stretch velocity. Golgi tendon organs are located in the junction between muscle and tendon, providing feedback on muscle force. Reflexes are involuntary responses to stimuli, and their strength is known to be modulated. Experiments have shown that different task instructions (Doemges and Rack 1992a, b; Abbink et al. 2004) , disturbance properties and dynamic properties of the environment ( Van der Helm et al. 2002) all elicit adaptation of the reflexive contribution to joint dynamics. Model simulations have indicated that these reflex settings are close to optimal for suppressing the disturbances (Schouten et al. 2001 ).. Additional to the many studies that quantify reflexive behavior by directly deriving metrics like integrated EMG after the application of a mechanical or electrical stimulus (e.g. Dewald and Schmit 2003; Schuurmans et al. 2009; Kurtzer et al. 2008; Nielsen et al. 2005) , other studies use a control engineering approach to parameterize joint dynamics. A combination of system identification and modeling is then used to separate muscular and reflexive contributions (Kearney et al. 1997; Perreault et al. 2000 ; Van der Helm et al. 2002; Ludvig and Kearney 2007; Schouten et al. 2008a ). The dynamic behavior of the joint is estimated by perturbing the joint with a random force disturbance using a robotic manipulandum and recording displacement and interaction force between the subject and the manipulandum. The joint dynamics are expressed in terms of the mechanical admittance, describing the amount of displacement per unit force. To separate reflexive contributions from muscle viscoelasticity, lumped parameters (Schouten et al. 2008b) can be fitted to the mechanical admittance. The resulting set of parameters quantify the joint inertia, the muscle viscoelasticity and the reflexive gains of the force, position and velocity feedback pathways.. There are multiple mechanisms that affect the strength of the reflex pathway. Fusimotor drive (Crowe and Matthews 1964; Ribot-Ciscar et al. 2009 ) modulates the muscle spindle's sensitivity to stretch and stretch velocity, therewith altering the gain of the reflex pathway. With presynaptic inhibition (Rudomin and Schmidt 1999; Rudomin 2009; Baudry et al. 2010) , the efficacy of transmission between primary afferent fibers and the receiving neurons can be centrally modulated. The strengths of interneuronal connections in the spinal cord affect the net amount of activation of the motoneuron pool due to the sensory feedback. All these mechanisms contribute to the joint dynamics as determined experimentally.. Despite the known neural mechanisms of reflex modulation and the available identification techniques, there are still many unknowns on how exactly neural mechanisms affect the measured joint dynamics. The goal of this study was to investigate how neural and sensory properties could map onto the (limited) set of reflex gains as obtained from joint dynamics, using a neuromusculoskeletal model. In other words: how do the underlying, physical neural and sensory properties effectively link to the lumped reflex gains? We used a neuromusculoskeletal model of a spinal neural network controlling an antagonistic pair of muscles with afferent feedback (Stienen et al. 2007) . In a sensitivity analysis, sensory and neural properties in the model were systematically varied and reflex experiments were simulated to determine reflex gains. The results showed that besides the expected mappings, there also exist intricate, counterintuitive relationships between neural and sensory properties and reflex gains that need to be taken into account when interpreting experimental results. Multiple mechanisms contribute to posture maintenance. Perturbation experiments and system identification are widely used to assess posture maintenance in humans (e.g. Kearney et al. 1997; Perreault et al. 2000; Van der Helm et al. 2002; Ludvig and Kearney 2007) . System identification and neuromuscular modeling pose a powerful tool for quantifying posture maintenance with a small set of control parameters. For instance, a position feedback gain expresses force buildup related to muscle stretch. However, many neural and sensory properties contribute to these feedback gains. When posture maintenance is captured in such a small set of reflex parameters, interaction could be expected. For instance, both decreasing presynaptic inhibition of the Ia afferent and increasing fusimotor drive can increase the velocity feedback gain, since the net amount of velocity related feedback in the control action to the muscle increases. These interactions complicate the interpretations of the results from reflex experiments. Neural and sensory mechanisms might affect multiple parameters of the lumped reflex model (see Figs. 5 and 6) . Since direct in vivo measurements of neural and sensory properties are unfeasible during posture control experiments in humans, the goal of this study was to investigate how neural and sensory properties could map onto the (limited) set of reflex gains. Or, in the example of presynaptic inhibition and fusimotor drive, investigate how the net synaptic weights between afferents and motoneurons or the parameters of the muscle spindle model affect the lumped reflex gains.. We simulated posture control experiments with a neuromusculoskeletal model. In the NMS model, spinal cord circuitry controlled a joint, while receiving feedback from muscle spindles and Golgi tendon organs in the activated muscles. The joint was perturbed with a random force disturbance. Like in experiments, the posture control behavior of the simulated NMS model was captured by fitting the parameters of a lumped reflex gain model onto the simulated data. The high VAF of the reflex gain model fits onto the output of the NMS model indicated that the linear reflex gain model was able to adequately capture the behavior of the strongly nonlinear NMS model. This was greatly due to the conditions in the simulated experiment (and commonly used in in vivo experiments): during a posture control task (minimize deviations), there is a constant level of muscle co-contraction with small deviations around the equilibrium position of the joint. Therefore, the experimental conditions linearize the behavior of the non-linear neural system. A comparison to actual experimental data showed that the reflex gains fitted from the neuromuscular model closely resembled reflex gains identified in vivo.. The results showed a dominant role of tonic descending excitation (TDE) on the parameters of the lumped reflex gain model. TDE to the motoneurons (parameter w td−MN ) increases tonic muscle activation (cocontraction) and therewith the muscle viscoelasticity, expressed in parameters b and k. Parameter w td−ALL simultaneously increased the amount of TDE to both motoneurons and all other populations receiving TDE. Besides viscoelasticity, also reflex gains k v and k f were influenced by these two TDE parameters; these reflex gains decreased with TDE. In the simulations of the NMS model, TDE increased activation of the receiving populations. As a result, the neural network received relatively less afferent input than tonic input, which could explain the decreasing reflex gains with TDE.. The results have shown that each of the lumped reflex gain parameters is positively sensitive to its most direct neural substrate: k v increases with parameters related to Ia afferent stretch velocity feedback pathway, k p with the II afferent stretch feedback pathway, and k f with Ib afferent force feedback pathway. For k v this most direct substrate was also the most dominant and interaction with other pathways was low. In contrast, k p and k f showed stronger sensitivities to other (and less directly related) neural and sensory mechanisms, which makes accurate identification difficult in the given experimental conditions. Both k p and k f decreased with velocity feedback over the Ia afferent pathway. Additionally, force feedback gain k f strongly decreased with TDE. This is in concordance with experimental findings: force feedback modulation in humans was strongest in an experimental condition with low co-contraction and low-frequent (slow) perturbations (Mugge et al. 2010) . Position feedback gain k p was remarkably less sensitive to neural and sensory property changes than k v and k f . In the posture control task used in the experiments deviations were small, while high muscle stretch velocities resulted from the disturbance force. Therefore muscle length information played a lesser role during the posture control task than velocity feedback. These results suggest that to effectively identify k p experimentally, stretch velocities need to be kept small to minimize the interactions between velocity and position feedback.. Summarizing, the sensitivity analysis demonstrated dominance of muscle spindle velocity feedback and tonic descending excitation. Velocity feedback gain k v was distinctly influenced by afferent velocity feedback and TDE, while position and force feedback gains k p and k f depended on a wide variety of neural properties. The results of this modeling study contribute to determining experimental conditions suited for studying reflex modulation in vivo. From reflex experiments Schouten et al. 2001) , we know that experimental conditions like level of cocontraction, perturbation velocities and amplitudes, but also task instruction induce reflex gain modulation. When the relationship between neural properties and reflex gains is better understood, experimental conditions can be adapted to the type of reflex modulation that is being studied. Furthermore, it is important to know if the experimentally estimated reflex gains are the result of the expected neural mechanism or may be a combination of interacting mechanisms. For example, one must be aware that modulation of k v can, besides Ia afferent feedback, be caused by tonic descending excitation to the motoneurons. It could therefore be important to control the level of co-contraction in such an experiment.. Like most models, the model presented here has many simplifications. Starting at the sensory level, a straightforward feline muscle spindle was chosen, together with a linear Golgi tendon organ model. Because of the modeled task (a position task with cocontraction and small, continuous perturbations with fixed frequency content) both sensors operate with small deviations around a relatively steady state. We argue that under these conditions, the simple models capture sufficient detail to describe the afferent feedback triggered by the perturbations, although we are aware of the intricate dynamics that muscle spindles and Golgi tendon organs can demonstrate . The sensory models used had their output in spikes per second and a Poisson process was used to convert output to spike trains. Halliday and Rosenberg (1999) presented a point-process spectral estimate on a human Ia afferent spike train, that closely matched the spectral properties of recorded spike trains, indicating that a point process could describe the Ia afferent dataset.. The spinal neural model was relatively straightforward. The spinal populations in the neural model were homogeneous, and connections between the individual neurons in the populations were assigned using a random process. The neuron model was a simple point neuron, without spatial representations like a dendrite structure. These are important simplifications, and more detailed models of the individual components of our neural model do exist. Nevertheless, this simplicity serves a purpose. The main goal was to determine how changes in velocity-, position-and force-related neural feedback map onto the limited set of reflex gains that are used in experiments. The main question was: which neural pathways could be contributing when changes in reflex gains are observed in an experiment? Unlike a lumped reflex gain model, where each feedback channel has a distinct pathway, biological afferent feedback fans out in the spinal populations before finally converging at the motoneurons. The multitude of pathways and their interactions are the cause of most lumped reflex gains being sensitive to multiple neural and sensory parameters. Each detail added to the neural model adds one or more parameters and thus might add a new relationship with one of the reflex gains. However, the main phenomenon that was demonstrated here can be captured with a relatively simple model that includes at least the afferent feedback pathways, fan-out and convergence in the spinal populations and output to a musculoskeletal model.. We conclude that with postural control tasks and wide-bandwidth force perturbations, parameters of the Ia afferent feedback pathway largely contribute to the task performance enabling accurate identification of k v . In these conditions of high stretch velocities, but low stretch and fairly constant muscle activation, parameters of the II and Ib afferent feedback pathways hardly contribute to task performance. Because of this low contribution to task performance, the amount of position and force related information in the measured joint deviations is low, resulting in poor identifiability of k p and k f . This is expressed by the high sensitivity to other, seemingly non-related feedback pathways. Identification of k p and k f probably needs an experimental design that decreases the effects of interaction from velocity feedback, like Mugge et al. (2010) suggests for k f . As an example, Fig. 4 shows how lumped reflex gains were modulated by varying neural parameter d Ia ; the velocity component of the muscle spindles. The sensitivity measure S ij is indicated in the figure. The example The sensitivity of the lumped reflex gains to variations in the neural and sensory parameters of the NMS model is illustrated in Fig. 5 . For each of the parameters of the lumped reflex gain model and the RMS of the joint deviation, the eight neural and sensory parameters with the largest effect (per parameter) are shown. The height of the bars shows the magnitude of the sensitivity metric S ij , plus and minus signs above the bars indicate the sign of S ij .. The top row in Fig. 5 shows the sensitivity of the intrinsic parameters: inertia m, joint viscosity b and joint stiffness k. Sensitivity of the inertia estimate is low for all parameters (sensitivity in the order of 0.05) as can be expected, since mass was not varied. Joint viscosity b is mainly affected by tonic descending excitation. Increasing TDE to specifically the motoneurons or to all populations increased viscosity by increasing the tonic muscle activation. Further, viscosity was increased by various excitatory pathways to the motoneurons and decreased by inhibitory pathways via the inhibitory interneuron populations. Sensitivity of stiffness k is remarkably low (in the same order of magnitude as that of inertia m). Because of the linearized muscle model, one might expect similar sensitivity of stiffness and viscosity. The difference is due to the relatively high value of stiffness k in the nominal condition, decreasing the normalized sensitivity.. The middle row in Fig. 5 illustrates the sensitivity of reflex gains k p , k v and k f to the eight parameters that they were most sensitive to. Velocity feedback over the Ia afferent decreased k p . This is indicated by the negative sensitivity to muscle spindle constant d Ia , and also by the positive sensitivity to inhibitory pathways from Ia afferents to motoneurons. Group II afferent feedback increased k p as expected. Renshaw cell activation increases position feedback gain k p , both through the long latency constant synapse (w MN−RC(L) ) and the synapse between the Renshaw cells and the IA interneuron (w RC−IA ). Velocity reflex gain k v increases with the stretch velocity components in the Ia afferent pathway (d Ia , e Ia ) and with increasing strength of the synapse between Ia afferent and motoneuron. TDE to either the motoneurons alone or to all populations decreases k v . Further, k v decreases with the stretch component of the Ia afferent, and the pathway over inhibitory interneurons. Activation by excitatory interneurons decreased k v , which might be caused by these interneurons receiving stretch feedback from the II afferent and no velocity feedback from the Ia afferent. Force feedback gain k f strongly decreased with TDE and Ia afferent feedback and increased with Golgi tendon organ (GTO) feedback over the Ib afferent pathway. The neural time delay of the Ib afferent increased k f as well.. The bottom row in Fig. 5 illustrates the sensitivities of neural time delay τ del , muscle activation time constant τ act and the RMS of joint deviation. Neural time delay estimate τ del was highly sensitive to the Ia afferent and efferent neural time delays as expected, and further to a mixture of neural and sensory parameters. The high sensitivity of the estimated muscle activation time constant τ act to a wide range of neural and sensory parameters is surprising (note that muscle activation was not varied in the simulations).. The bottom-right panel in Fig. 5 shows how the RMS of the joint deviations changed with the neural parameters. TDE had the strongest effect on performance, by increasing motoneuron activity and therewith cocontraction. Further, Ia afferent feedback (both synaptic strength and muscle spindle properties) improved performance while neural time delays decreased performance. Force feedback over Ib afferents and position feedback over II afferents only had a small effect on performance. Figure 5 demonstrates that most lumped reflex gains were sensitive to a mixture of neural and sensory parameters. To elucidate the relation between the properties of the proprioceptors and the estimated reflex gains, Fig. 6 shows a subset of the data: the sensitivity of k p , k v and k f to only the sensory constants of the muscle spindle and GTO. The velocity components of the muscle spindle (constants d Ia and e Ia ) positively correlated with velocity gain k v , with relatively low interaction with the other reflex gains. Position feedback gain k p and force feedback gain k f however did not show such a distinct sensitivity. There was positive sensitivity of k p to the stretch component of the muscle spindle c II , but k p decreased with velocity component d Ia as well. The GTO constant c Ib led to an increase of k f , but the spindle parameters a Ia , c Ia and d Ia have a far stronger negative (decreasing) effect on k f . Summarizing, Fig. 6 demonstrates that k v and k f are mostly influenced by muscle spindle feedback, while k p is influenced by a mixture of afferent feedback pathways. Table 1 for a list of these parameters)

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Probabilistic polynomial dynamical systems for reverse engineering of gene regulatory networks Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for un-certainty, and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms. The enormous accumulation of experimental data on the activities of the living cell has triggered an increasing interest in uncovering the biological networks behind the observed data. This interest could be in identifying either the static network, which is usually a labeled directed graph describing how the different components of the network are wired together, or the dynamic network, which describes how the different components of the network influence each other. Identifying dynamic models for gene regulatory networks from transcriptome data is the topic of numerous published articles, and methods have been proposed within different computational frameworks, such as continuous models using differential equations [1, 2] , discrete models using Boolean networks [3] , Petri nets [4] [5] [6] , or Logical models [7, 8] , and statistical models using dynamic Baysein networks [9, 10] , among many other methods. For an up-to-date review of the state-of-the-art of the field, see, for example [11, 12] . Most of these methods identify a particular model of the network which could be deterministic or stochastic. Due to the fact that the experimental data are typically noisy and of limited amount and that gene regulatory networks are believed to be stochastic, regardless of the used framework, stochastic models seem a natural choice [9, 13, 14] . Furthermore, discrete models where a gene could be in one of a finite number of states are more intuitive, phenomenological descriptions of gene regulatory networks and, at the same time, do not require much data to build. These models could actually be more suitable, especially for large networks [15] .. The discrete modeling framework for gene regulatory networks that has received the most attention is Boolean networks, which was introduced by Kauffman [3] . They have been used successfully in modeling gene regulatory and signaling networks; see, for example [16] [17] [18] . Many reverse engineering methods have been developed to infer such networks, see, for example [19, 20] .. For the purpose of better handling noisy data and the uncertainty in model selection, Boolean networks were extended to probabilistic Boolean networks (PBN) in [13, 21, 22] . A PBN is a Boolean network where each node i may possibly have more than one Boolean transition function, say f i1 , . . . , f it i , where t i ≥ 1, and, to decide the future state of i, a function f (i) j is chosen with probability p ij , where p i1 + · · · + p it i = 1. To be precise, to each node i in a PBN, the set F i = {(f ij , p ij )} j=1,...,t i of possible transition functions and their probabilities is assigned. Notice that if t i = 1 for all nodes in the network, then the PBN is just a Boolean network. As it is the case with Boolean networks, a PBN could be updated synchronously or asynchronously. However, throughout this article, we focus on synchronous PBN. Aspects of PBNs, and also asynchronous PBNs, have been studied in, for instance [23, 24] and they have been applied to the modeling of gene regulatory networks in, for example, [25, 26] . Furthermore, methods for inferring PBN have been developed in [27] .. One disadvantage of Boolean models for gene regulatory networks is the limited number of states in which a gene can be. Indeed, although for a molecular biologist the state of a gene is usually discrete, it could be not only "expressed" and "not expressed" but also "over expressed," for example. There has thus been some consideration of more-than-binary discrete models in the Boolean network community. In the context of PBNs, generalizations of Boolean networks for ternary gene expression have been proposed in [28] [29] [30] [31] . In addition, in [32] a ternary model has been considered as a preliminary stage for a Boolean one.. Other discrete multistate modeling frameworks have been developed too. Logical models [8] and K-bounded Petri nets [6, 33] are two multistate modeling frameworks that have been used for modeling gene regulatory networks. A natural generalization of Boolean networks to multistate networks are the so-called polynomial dynamical systems (also known as algebraic models), which were introduced in [34] . In an algebraic model, the set of possible states of each node is a finite set, and once the mathematical structure of finite fields is imposed on that set, the transition function of each node is necessarily a polynomial. As this framework is rooted in computational algebra and algebraic geometry, results from these fields are used for the reverse engineering of dynamic and static biological networks [34] [35] [36] [37] , as well as for analyzing model dynamics [34, 38] , which usually is a challenge. Furthermore, in [39] , it was shown that logical models and K-bounded Petri nets can be viewed as polynomial dynamical systems and algorithms for their translation into algebraic models were provided which facilitates the analysis of their dynamics.. In this article, we first introduce a stochastic generalization of polynomial dynamical systems, namely, probabilistic polynomial dynamical systems, which is also a generalization of the above-mentioned probabilistic Boolean networks to multistate models. Then, using this framework, we present a novel method for the reverse engineering of multistate gene regulatory networks from limited and noisy data. The novelty of our approach is two-fold. First, the stochastic model we construct is based on all minimal models in the model space and second, the probabilities assigned to the minimal models are based on an algebraic partition, called Gröbner fan, of the models space, which provides an algorithmic and algebraic method for the construction of such stochastic models.. In the next section, we present our method for the reverse engineering of gene regulatory networks as probabilistic polynomial dynamical systems. Then we demonstrate this method using the yeast cell cycle model in [17] , as well as the synthetic network of the yeast cell cycle in [40] . Gene regulatory networks are structured as inter-connected entities and their complex nature is inherently stochastic. The framework of stochastic dynamical systems is natural for modeling and analyzing such networks. We focused on PPDSs due to their applicability to limited and possibly noisy data. Within this modeling framework, we developed a systematic method based on combinatorial topology, algebraic geometry, and statistics for the reverse engineering of the dynamics, as well as the gene dependencies, in biochemical regulatory networks from experimental data. The algorithm can handle large regulatory networks and hence is applicable to many networks of interest. The constructed models are comprised of minimal polynomials according to the definition in [34] . We plan to explore the use of other types of biologically relevant functions, such as nested canalyzing functions [53] . An algorithm for the inference of deterministic nested Boolean canalyzing networks has recently been presented (F Hinkelmann, A Jarrah: Inferring biologically relevant models: nested canalyzing functions, submitted). Combining this with our algorithm here will provide a systematic method for the reverse engineering of gene regulatory networks as probabilistic Boolean nested canalyzing networks.

O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O