Spiking Neural Networks constitute the most promising approach to develop realistic Artificial Neural Networks (ANNs). Unlike traditional firing rate-based paradigms, information coding in spiking models is based on the precise timing of individual spikes. It has been demonstrated that spiking ANNs can be successfully and efficiently applied to multiple realistic problems solvable with traditional strategies . In recent years, major breakthroughs in neuroscience research have discovered new relevant computational principles in different living neural systems. Could ANNs benefit from some of these recent findings providing novel elements of inspiration? This is an intriguing question for the research community and the development of spiking ANNs including novel bio-inspired information coding and processing strategies is gaining attention. From this perspective, in this work, we adapt the core concepts of the recently proposed Signature Neural Network paradigm—i.e., neural signatures to identify each unit in the network, local information contextualization during the processing, and multicoding strategies for information propagation regarding the origin and the content of the data—to be employed in a spiking neural network. To the best of our knowledge, none of these mechanisms have been used yet in the context of ANNs of spiking neurons. This paper provides a proof-of-concept for their applicability in such networks. Computer simulations show that a simple network model like the discussed here exhibits complex self-organizing properties. The combination of multiple simultaneous encoding schemes allows the network to generate coexisting spatio-temporal patterns of activity encoding information in different spatio-temporal spaces. As a function of the network and/or intra-unit parameters shaping the corresponding encoding modality, different forms of competition among the evoked patterns can emerge even in the absence of inhibitory connections. These parameters also modulate the memory capabilities of the network. The dynamical modes observed in the different informational dimensions in a given moment are independent and they only depend on the parameters shaping the information processing in this dimension. In view of these results, we argue that plasticity mechanisms inside individual cells and multicoding strategies can provide additional computational properties to spiking neural networks, which could enhance their capacity and performance in a wide variety of real-world tasks. Biological neural circuits are powerful computational systems that efficiently process a great amount of data in real time with extensive plasticity capabilities. This makes the nervous system a source of inspiration when designing engineered tools. In this sense, many Artificial Neural Network (ANN) paradigms mimicking the computational principles performed by living neural systems have been developed to solve real-world problems  in the nervous system seems to be apparent  is a cell-specific intraburst spike timing distribution described as the sequence Si = {ISI1, ISI2, …, ISIn}, where ISIn represents interspike intervals between consecutive spikes within the same burst. The timing of the last spikes in the bursting activity of the pyloric CPG cells varies from one burst to another; while the first spikes in the burst are highly reliable . The spike timings of the second part of the burst are given by a preferred output pattern  that changes dynamically as a result of the single neuron plasticity (see Section 2.1.3).Signature neural networks use neural fingerprints to identify each individual unit of the ensemble . Therefore, different spatio-temporal spaces can be simultaneously used to globally encode and store information. In the network discussed in this paper, we consider two coexisting units of information in each neural signal: the bursting frequency and the neural fingerprints included within the burst. In the first dimension, the network must generate and coordinate spatio-temporal patterns of propagating transient bursting activity . To achieve this, we impose two constraints , its “membrane potential.” Figure p—the transit probability of the internal state per time step. When the membrane potential of a neuron ni reaches the firing threshold (TH), this generates a sequence of spikes (not a single spike). The temporal distribution of spikes within the response burst is given by a firing sequence composed of concatenating the signature (Si) and the preferred output spike pattern (Pi) of the neuron. Then, the stochastic dynamics of a single neuron depends on the temporal evolution of the neuron activity and whether it is under (subthreshold activity) or over (spiking-bursting activity) the firing threshold. Formally:To describe the individual behavior of each unit, we define a stochastic model operating in a discrete event framework. The neuron activity is considered as a discrete variable and characterized in time by Vi(t) < TH):During subthreshold activity (Isyn is the synaptic input (Equation 3) and p the transit probability of the internal state per time step.where Vi(t) > = TH):During the generation of the burst (N is the number of spikes in the firing sequence (Si + Pi), tn denotes the timing of the nth spike in this sequence,  and being AP the peak membrane potential to generate a spike. Note, that during the burst generation synaptic input (Isyn) is not taken into account (cf. Equation 1 and 2). After generating a burst, neurons have a refractory period of RP time steps during which Vi(t) = 0. Then, subthreshold dynamics starts again.where Synaptic input arrives to a neuron through two kind of input channels: connections with other neurons and an external channel to introduce external stimulation into the network. Each neuron in the network is connected to its eight nearest neighbors Figure  with perge defines the weight of the external stimulus, pulsee is 1 when an action potential is delivered through the external channel and 0 otherwise; and, similarly, gji is the weight of the connection between neurons nj and ni and pulsej is 1 when Vj(t − 1) = AP and 0 otherwise. Note that Equation 3 does not apply neither during the generation of a burst (Equation 2) nor during the refractory period, i.e., in these situations synaptic input is not considered.where gji is constant for all the synapses and, consequently, the neighborhood of every neuron does not change.It is important to highlight that in this paper we do not discuss synaptic learning (see Section 4). This implies that Incorporating subcellular plasticity to a neuron model implies that a mechanism inside the cell allows tuning the neuron dynamics to incoming signals and/or to particular processing states. We consider here a history-dependent contextualization mechanism driving the spike-timing encoding modality. This intra-unit contextualization modulates the preferred output pattern as a function of previous incoming spike patterns.local informational context. The local informational context keeps track of the information received during a time window of Mi time units, providing a history-dependent contextualization mechanism to the single neuron processing. In the case of our spiking network, for each incoming spike, the neuron stores in its local context the joint information about the input channel and the spike timing . At time step 136 (pointed by the arrow), an spike arrives through channel C1. The neuron can use its local informational context . The neuron emits the new preferred output pattern until a new fingerprint is recognized or until the recognized fingerprint appears less than Li times in the local informational context (keep in mind that this is transient memory). Note, that intra-unit plasticity can be used to compute different aspect of the output signal as a function of the local contextualization, not only the spiking firing pattern. For instance, a particular cell could increase/decrease its level of activity or generate an output spike in response to specific incoming patterns independently of the synaptic weight.Figures t Figure  to conteDuring the input processing, channels are checked randomly in each iteration. In this way, when the target neuron recognizes multiple signatures in the same iteration, the last processed prevails over the others. Plasticity rule does not apply during the generation of a burst—i.e., once the neuron starts firing, the output spike pattern cannot change.Vi(t)) is represented with a color scale. Regions with the same color have synchronous behavior. Red corresponds to neurons with a membrane potential over the firing threshold (Vi(t) > TH), i.e., they are generating a burst. Intermediate colors between blue and red, represent subthreshold activity. The cooler the color, the lower the level of activity.To illustrate the spatio-temporal patterns generated in the bursting informational dimension, we generate activity movies representing the membrane potential evolving dynamics. In these movies, the evolution in time of the activity of a given unit , provides an estimation of the complexity of the image corresponding to the spatio-temporal pattern at time t. A low number of coefficients means that the image is smooth or is composed of smooth components. In contrast, a high number of coefficients corresponds to complex images. In this way, the DWT analysis transforms the multidimensional spiking-bursting activity in the network, Vi(t), into a one dimensional signal, C(t). This signal provides an useful characterization of the bursting dynamics in which both the frequency and the spatial complexity can be discussed. From the frequency perspective, a simple visual inspection of the evolution of C(t) allows to detect the presence of different rhythmic patterns in the network. Furthermore, these rhythms can now be studied by means of the one dimensional Fourier transform. From the spatial complexity of the patterns, very high values of C(t) correspond to almost random behavior of every neuron, with no patterns present; intermediate high values indicate the presence of complex spatial structures in the patterns; while completely synchronized networks produce a small number of coefficients. Note, that C(t) ranges between 0 and the number of neurons in the network.In particular, we use the same discrete wavelet transform (DWT) analysis employed in Latorre et al.  to charaFigure 5). Each point in the 50 × 50 square represents with a color code the neural signature recognized by a given neuron within the network at a given moment. In this manner, neurons with the same color recognize the same signature. White color identifies the units that do not recognize any fingerprint.The spike-timing encoding is related to the spreading of specific intraburst spike patterns through the network and the synchronization mechanisms that allow a group of neurons to recognize and emit the same signature at a given moment , the firing threshold (TH), and the duration of the refactory period (RP). These parameters modulate the ratio of bursts produced by an isolated neuron. The greater the value of the stochastic probability p, the higher the mean bursting frequency. Similarly, the bursting frequency also grows with low values of TH and RP.We have conducted experiments in which multiple datasets are presented to regular networks with different parameters. Independently of the network size and the number of neighbors per neuron, it is possible to find a broad range of synaptic weights and neuron parameters allowing the network to simultaneously encode information in the rhythmic and the spike-timing modality. However, the emerging phenomena that we describe here can be more easily illustrated in autonomous networks with a low level of bursting activity, since in these cases, the spatio-temporal activity in the different dimensions arises due to external stimulation. In autonomous networks, i.e., networks not receiving external input, the level of bursting activity depends on the transit probability of the internal state . Note, that AP—the peak membrane potential to generate a spike—has no influence on information processing, the only requirement is being greater than TH. We discuss results of square-shaped networks of 50 × 50 of such units, with periodic boundary conditions and where each unit is connected through an excitatory synapse (gji = 1) to its eight nearest neighbors as shown Figure Vi(0) is chosen randomly between 0 and 40 a.u. for all neurons in the network.Thus, in the following sections, we focus on neurons where p and TH, the degree of synchrony varies as a function of the synaptic transmission strength among neurons . For small values, each neuron fires nearly independently. As synaptic weights grow, the degree of synchrony increases because the generation of a burst in a given unit sequentially propagates to its neighbors and so on . However, as we mention above, here we are interested in autonomous with a low level of bursting activity.The degree of synchrony among the membrane potential of the neurons constituting the network characterizes the global spiking-bursting activity in the network. For fixed values of n Figure . The higDepending on the synaptic parameters, burst propagation provides autonomous networks the ability to generate well-defined spatio-temporal patterns in the form of propagating wave fronts of transient spiking-bursting activity. Note that local contextualization modulates intraburst firing patterns, but it has not any influence on burst timings. To illustrate these spatio-temporal patterns, we generate activity movies representing the membrane potential evolving dynamics (see Section 2.2.1 for details). As representative example of the spontaneous collective bursting rhythms generated by the network, bottom panel in Figure The spontaneous generation of transient spatio-temporal patterns of spiking or spiking-bursting activity is a feature with relevant functional implications observed in different living neural media. However, we are interested in the network response to stimuli. Therefore, from the encoding perspective, the most interesting feature of the network, appearing even in networks with a small synaptic transmission among neurons, is its ability to develop dynamical patterns of spiking-bursting activity in response to data onset. These patterns allow the network to encode information using the frequency of different bursting rhythms induced by stimuli. To illustrate how the network of Figure p and can be estimated by means of the Fourier transform of the wavelet analysis of the network activity. For instance, in the network of Figures −3 . The frequency of the spiking-bursting rhythms evoked by external stimulation depends on the frequency of the input, since the stimulated neuron follows the stimulus. These changes indicate that the network has encoded the incoming information in a characteristic spiking-bursting rhythm. Finally, no input is present again and the network recovers the spiking-bursting autonomous activity  corroborates the rhythm encoding in the network transient spiking-bursting dynamics. Changes in the collective spiking-bursting dynamics in response to data onset are reflected in a change in the evolution of the DWT coefficients whose shape characterizes the spiking-bursting activity of the network. As an example, Figure .2.1 corrThe emerging collective dynamics analysis in networks that receive multiple tonic stimuli with different frequencies indicate that spatio-temporal patterns of spiking-bursting activity allow the network to encode information using several coexisting and coordinated rhythms. Top panel in Figure p, in this situation, the network even becomes a long-term memory of spiking-bursting rhythms. We would like to emphasize that the rhythms that survive for longer periods in short-term memories or the ones that persistently reverberate in long-term memories are not always the higher frequency stimuli-evoked rhythms nor the rhythms encoding the last data presented to the network.We have previously shown that the spiking-bursting rhythms evoked by a single stimulus reverberate for a while when the stimulation is over. The reverberation period drastically increases when the network receives multiple stimuli. The greater the number of external inputs, the greater the number of sources of spiking-bursting activity. This translates into a higher spiking-bursting activity in the network and explains the increasing reverberation period. Depending on the synaptic strength and the value of One of the major characteristics of the proposed network is the intra-unit contextualization of input signals, responsible of the spike-timing encoding modality. In this section, we study the complex collective dynamics induced by this intra-unit information processing strategy. These emerging collective dynamics can give us important clues about the underlying computational properties of the network.Mi) and the fingerprint learning threshold (Li). Depending on the value of these parameters, specific intraburst firing patterns can propagate through autonomous networks. However, the more interesting phenomena from the information processing viewpoint are related to the mechanisms that allow the network to generate and organize spatio-temporal patterns in response to data onset. Therefore, we focus our attention on networks in which the signature recognition does not occur without external stimuli. When these networks receive incoming data, they aid the study of the information encoding in the fingerprint-based spatio-temporal space by analyzing how the signatures of the stimulated units propagate throughout the network.As expected, the ability of an individual unit to recognize specific fingerprints varies as a function of the intra-unit parameters shaping local contextualization, i.e., the maximum size of the local informational context  in a distributed network form.Again, we first address the analysis of networks receiving a single stimulus. When a neuron receives an external tonic input, this unit increases its bursting frequency (see Section 3.1). This increase can make the neighbor units recognize the neural signature of the stimulated neuron and propagate the corresponding intraburst firing pattern. In this situation, new intriguing collective dynamics arise in the network. To illustrate the dynamic spatial organization of the neural signatures traveling through the network, we generate activity movies representing the fingerprint-based evolving dynamics (see Section 2.2.2 for details). These activity movies point out that the network generates in this dimension well-defined transient patterns of activity in response to data onset. The emerging spatio-temporal patterns are related to the spatial organization and clusterization of the signatures traveling through the network. To give insight into the generation and propagation of these complex spatio-temporal structures, Figure a.u). Likewise, the spatial organization of the patterns in the different spatio-temporal spaces is not correlated. If we consider that at a given moment two neurons over the firing threshold belong to the same cluster, we can calculate the clustering coefficient and the average shortest path for the spiking-bursting patterns and compare the self-organizing properties of the patterns encoded in both information dimension. This analysis points out that the spiking-bursting patterns always consist of propagating transient wave fronts from the stimulated unit traveling through the whole network. Meanwhile, the fingerprint-based patterns can also be originated in the stimulated unit, but they can propagate locally or globally as transient wave fronts or remain bounded in specific regions of the network.The information encoded in the spike-timing modality and the encoded in the rhythmic modality coexist in the network. A relevant property observed in the simulations is that a neural fingerprint does not necessarily travel over the propagating wave fronts encoding the corresponding spiking-bursting rhythm. The spreading velocity of the fingerprint-based spatio-temporal patterns is always slower than the corresponding spiking-bursting spatio-temporal patterns velocity  decreases the number of neurons that recognize a given firing pattern. On the other hand, decreasing the learning threshold (Li) facilitates the recognition of the propagating fingerprints and, therefore, the level of activity in the network grows. The trade-off among the effect of these parameters determines if the network encodes information in the spike-timing modality and the mode of behavior in this dimension. To illustrate this, Figure The collective dynamics in the fingerprint-based dimension is mainly driven by the intra-unit parameters With the experiments described so far, we investigate the ability of the proposed network to encode and process a single stimulus using an information processing strategy driven by local contextualization. If we repeat the same experiments but now introducing multiple inputs simultaneously, we observe that the presence of multiple stimuli makes the network generate coexisting transient spatio-temporal patterns of activity encoding the origin of the different inputs Figure . These eWe would like to highlight that the competition regimes observed in the activity movies arise in the absence of inhibitory connections, which hints at intra-unit contextualization as an effective mechanism to restrict the activity in networks without inhibition. Note that each neuron can only transmit one recognized firing pattern per burst. This limitation produces somehow a local competition among the patterns received by the neuron where only the “winner” is transmitted. This local competition is the basis of the global competition in the whole network.Mi and Li and may vary from a winnerless (WLC) to a winner-take-all (WTA) competition. In WLC networks, none of the signatures becomes a “winner,” and therefore, none of them persistently prevails over the others. Depending on the intra-unit parameters, the network can display different winnerless regimes. Figure The different dynamical modes observed in the activity movies are better characterized by the evolution of the number of neurons that recognize and emit each signature Figure . RegardlMi and Li, this can also spread as evolving transient patterns equivalent to the shown in Figure An interesting phenomenon observed with some network settings is that some regions within the network specialize in the emission of firing patterns encoding the origin of different stimuli although they do not receive any external input. This phenomenon occurs without any kind of supervised synaptic nor intra-cellular learning, i.e., it is a self-organizing property of the network. These emitter areas are usually related to winnerless competitions where the prevailing fingerprints change accordingly to the patterns originated in these areas Figure . ConversThe reverberating spatio-temporal patterns encoding the origin of incoming data continue competing even when they are not sustained by an active input. Short-term memory networks have a limited ability to retain previously stored data when new information is introduced into the network. In these cases, the reverberation period drops as compared to networks receiving a single input, and the stored data are almost instantaneously forgotten, i.e., the corresponding patterns disappear because the patterns encoding the last incoming stimulus win the competition. However, in long-term memory networks, coexisting coherent spatio-temporal patterns related to multiple fingerprints can be observed even when the corresponding input is not active.The present work introduces a spiking neural network that makes use of multicoding strategies for information propagation and subcellular plasticity to locally contextualize or discriminate data received by a unit. Furthermore, each neuron in the network has a neural signature that allows its unequivocal identification by the rest of the cells. This network is an encoder and generator of spatio-temporal patterns that take advantage of the multiple simultaneous encoding modalities present in the network to transform dynamic inputs into different spatio-temporal spaces, and organize and coordinate coexisting patterns of transient activity in response to data onset.The discussed experiments are aimed at analyzing the emerging collective dynamics in two information dimensions. On one hand, a spiking-bursting spatio-temporal space, where information processing is driven by synaptic transmission. On the other hand, a fingerprint-based spatio-temporal space driven by an intra-unit contextualization mechanism. The specific properties of the dynamic organization of the patterns are different in each information dimension, so that, the life cycle of the information encoded in both encoding schemes is independent. When multiple patterns in the same dimension coexist in the network, a competition emerges between them. We show that various forms of competition can arise without inhibitory connections in the network. Depending on the parameters shaping simple intra-unit plasticity rules, the competition regime may vary from a winnerless  to a winner-take-all competition . The stimuli-evoked spatio-temporal patterns and the corresponding competing dynamics can survive for long periods after data onset. This reverberation effect allows the network to memorize incoming data. This can display short-term or long-term memory capabilities in the different spatio-temporal spaces. When the network behaves as a short-term memory, the spatio-temporal patterns encoding incoming data in the corresponding scheme transiently reverberate after the stimulation ending. Conversely, in long-term memories, the stimulus leads the network to a new stable state and the patterns persistently survive. The memory ability of the network in each dimension varies as a function of the synaptic and/or intra-unit parameters. Therefore, different simultaneous processing strategies can be implemented within the network.These results illustrate the dynamical richness and large flexibility of the proposed network to encode and process information in different spatio-temporal spaces. We argue that plasticity mechanisms inside individual cells and multicoding strategies can provide additional computational properties to spiking neural networks, which could enhance their capacity and performance. In particular, local contextualization mechanisms allow individual neurons to process the multiple simultaneous codes in their input signals selectively or globally in order to completely decide or weight the decision about their output in the different encoding schemes. This information processing provides a framework to model complex high-dimensional processes that can be applied to different real-world computational problems. The ideas relating multicoding with local information discrimination have a direct application in problems that benefit from multifunctionality and parallelism. These are desirable features for many technical applications of ANNs, representing a potential advantage when processing large amounts of data or multiple decision-making criteria must be developed, for instance, in multiobjective optimization problems ; higher levels of bursting activity; and different number and/or distribution of spikes in the neural signatures are equivalent to the results presented in Section 3. In experiments with signatures with an arbitrary number of spikes, new interesting fingerprint-based dynamics emerges in the network and results are not exactly the same. In these simulations, not only the fingerprints belonging to a neuron propagate, but also specific firing sequences built with combinations of these signatures propagate throughout the network. In some sense, these networks do not only encode information regarding the input source, but they also generate new information. It is also important to note that, for simplicity, we only consider two encoding schemes in the network. However, bursting activity allows easily including additional units of information . In this line, and regarding a selective processing of input messages, experimental evidence indicates that some neural systems exhibit Although not addressed in this paper, subcellular plasticity and multicoding mechanisms for information processing can be combined with the features that underlie information processing in the existing spiking neural network paradigms. In this line, for example, plenty of work has been done on synaptic plasticity in spiking neural networks, since modifications of the synaptic connections are traditionally considered the physiological basis of learning in the nervous system. These works are mostly related to unsupervised synaptic learning methods, such as Spike-Timing Dependent Plasticity (STDP)  and MINECO/FEDER DPI2015-65833-P.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The use of antenna arrays in Global Navigation Satellite System (GNSS) applications is gaining significant attention due to its superior capability to suppress both narrowband and wideband interference. However, the phase distortions resulting from array processing may limit the applicability of these methods for high precision applications using carrier phase based positioning techniques. This paper studies the phase distortions occurring with the adaptive blind beamforming method in which satellite angle of arrival (AoA) information is not employed in the optimization problem. To cater to non-stationary interference scenarios, the array weights of the adaptive beamformer are continuously updated. The effects of these continuous updates on the tracking parameters of a GNSS receiver are analyzed. The second part of this paper focuses on reducing the phase distortions during the blind beamforming process in order to allow the receiver to perform carrier phase based positioning by applying a constraint on the structure of the array configuration and by compensating the array uncertainties. Limitations of the previous methods are studied and a new method is proposed that keeps the simplicity of the blind beamformer structure and, at the same time, reduces tracking degradations while achieving millimetre level positioning accuracy in interference environments. To verify the applicability of the proposed method and analyze the degradations, array signals corresponding to the GPS L1 band are generated using a combination of hardware and software simulators. Furthermore, the amount of degradation and performance of the proposed method under different conditions are evaluated based on Monte Carlo simulations. Jamming signals can significantly degrade the performance of receivers or completely deny GNSS position and time services. Many applications primarily relying on GNSS are vulnerable to in-band interference. Even a low-power interference signal can potentially jeopardize GNSS receiver’s performance in a circular region around the source within a radius of several kilometres. Although laws prohibit usage of such devices, low-cost jammers popularly known as personal privacy devices (PPD) can be legally or illegally procured. Spatial processing techniques utilizing an array of antennas can effectively defeat interfering signals regardless of their temporal or spectral characteristics; they surpass time and frequency domain processing methods due to their capability to suppress wideband interference. As long as the cost and power consumption of additional front-ends and antennas are justifiable, spatial processing is one of the most powerful countermeasures against various types of interference and jamming signals.®) [Beamformers and antenna arrays have been considered effective for a variety of signal processing applications such as radar, wireless communications and speech enhancement. Antenna array processing in GNSS applications has been mostly centered on interference suppression ,2,3,4. G®)  is an exResearch on GNSS anti-interference mostly focuses on interference suppression capability and there is less research on the distortion and degradation caused due to the spatial processing itself, particularly for high accuracy applications. Research on interference impact on GNSS receiver parameters has been mostly focused on single antennas ,12,13. IDistortion due to the space-only processing method has comparatively received less attention in the literature. Array-based receivers performing space-only processing may experience phase distortions, which in turn degrades the position and time estimates ,19,20,21This paper focuses specifically on the distortion caused on receiver parameters during blind beamforming. In non-stationary interference scenarios, the array weights should be continuously estimated and updated. Use of a blind beamformer for such scenarios may cause carrier phase degradations that are evaluated in this paper. The degradations are shown for GPS L1 array signals generated using a combination of hardware and software simulators. Moreover, the amount of degradation in different conditions obtained by Monte Carlo simulations is evaluated. A strategy to avoid the distortion on receiver parameters due to the blind beamforming process is also evaluated. Moelker  shows thMoreover, in all of the methods discussed above, the effect of carrier phase distortion on the receiver position solution has been not evaluated. In this paper, a new method will be devised to keep the simplicity of the blind beamformer structure and at the same time reduce tracking degradations. It is also shown that integer cycle carrier phase ambiguity resolution can be achieved when employing this method in interference environments. This is critical in high precision applications. For example, the base station and the rover receivers in a real time kinematic (RTK) network could be equipped with this beamformer to maintain sub-centimetre level accuracy while suppressing jamming signals. In the scope of this paper, the performance of the proposed blind beamformer is compared against conventional blind beamforming  for carrier phase based positioning.Throughout this paper, the following notation is adopted: small bold letters stand for vectors and capital bold letters stand for matrices. Superscripts N-element array configuration for the satellite signal and K interfering signals can be written asv is a vector specifying K interfering signal waveforms. C model the difference in amplitudes and phases due to unequal cable lengths and other electronic parts and its off-diagonal elements represent the coupling coefficients between antennas. In Equation (1), a is an n = 1, 2, …, N is a nth antenna element and ith), there is a corresponding element (jth) such thatSymmetric array configuration: For each antenna element (i and j) are also constrained to be conjugate of each other, such thatArray weights  and (7) and if it is assumed that the array is fully calibrated, then In order to estimate re known ,26. MoreIn this section, by taking advantage of the special array configuration, a phase compensation approach is proposed to compensate for beamforming phase changes. For this approach, the array needs not be fully calibrated and knowledge of array platform attitude parameters and the direction of the incident signals are not required. Moreover, the following estimation process is independent of the receiver structure. The compensation unit can be implemented within the structure of the conventional blind beamformer without feedback from the GNSS receiver (as shown in Considering Equations (8) and (9), a compensation matrix is applied to the received signal vector to satisfy the following condition:In order to estimate It is easy to conclude that the relation in Equation (10) holds ifMatrices and vectors in Equation (12) can be rearranged asEquation (13) can be written asThe compensation unit is independent of the null steering and can be considered as a parallel unit in the pre-despreading beamformer. It should however be noted that compared to the conventional blind beamformer, the degree of freedom of the array is reduced by half by employing this approach. This is a trade off between the number of interfering signals that can be suppressed and the accuracy of the position estimates obtained after mitigation.In the last step, the array gain vector to nullify interference can be obtained based on the estimated compensation matrix in a constrained optimization. Minimizing the output of the beamformer leads to the following optimization:The first constraint avoids the trivial solution which is Considering the symmetry in the array, and Equations (7) and (12), it can be easily concluded that The solution for this minimization is straightforward by using a Lagrange multiplier approach. Another approach is directly solving the optimization in Equation (17) similar to the one performed in . In the Incorporating the simulation of perturbations in array element positions or modifying the array miscalibration errors and generating GNSS signals with different signal settings is a time consuming task. Therefore, a Monte Carlo approach, which is an alternative and comparatively simpler way to evaluate the performance of the proposed method in different scenarios, is presented in this section; in the next section the effectiveness of the approach in a receiver and position domain is verified for one set of GPS array signals. The parameter settings used in Monte Carlo approach are given in Unequal cable lengths, mutual coupling among array elements, antenna phase and gain mismatches, and phase centre variations can affect the resulting phase and amplitude of the received signal at each antenna. In , it was The antenna gain patterns considered in the simulations correspond to the Novatel GPS Antenna 501 . In CaseAs discussed, in addition to the array gain, there is an induced phase due to the beamforming process that should be avoided in order to have undistorted phase from a beamformer. In the following simulations, the amount of this induced phase is evaluated. Additionally, for the sake of simplicity in illustrations, only induced phase due to the beamformer is considered and the phase of satellite signal is assumed zero. Therefore, any phase distortion appears in the form of a non-zero phase and it consists of an imaginary part.In the first simulation, the induced phase is illustrated  in real-Generally, even though the array configuration is chosen in advance, the positions of the antenna array element’s phase centres are not precisely known due to physical perturbation, phase centre variations and other uncertainties. This gives rise to antenna element position perturbations. These perturbations lead to deviation from the symmetric assumption in the array configuration. For the purpose of this analysis, perturbations in the range of 0 to 3 cm are considered. Perturbations of the chosen magnitude were added to each direction/co-ordinate . The Monte Carlo simulations for an eight element symmetric array were done using the settings provided in A higher value of the ratio metric indicates that the output has higher contributions from the real values compared to the imaginary values. From Performance of any beamforming differs for different numbers of antenna array elements. The performance of the method is now evaluated for different number of elements; however, a symmetric configuration is maintained as shown in Herein, GNSS array data are simulated using the recorded GPS digital baseband signals generated with a hardware simulator. Precise array signals are simulated using the simulation method outlined in . The dig0 for the conventional (phase uncompensated) method and the new approach (phase compensated) are compared. In the conventional beamformer, there is no assumption on the array configuration or having symmetric conjugate gain vector. Moreover, the compensation matrix is not applied to the received signal vector. Results for three PRNs are shown in 0 values show no attenuations. As observed in 0 values for both approaches are high and almost the same.The analysis was conducted using a multi-antenna GNSS software receiver that was developed starting from the single antenna GSNRx software receiver . The genSome preliminary experiments analyzing the accuracy of code phase based positioning indicated marginal improvements in some cases and significant improvements in others, depending on the interference scenario. To the contrary, significant improvements were always seen in the position results derived using the carrier phase measurements, irrespective of the interference scenario. Further analyses on the code phase measurements are required to assess the improvements obtained from the proposed method. The scope of this paper is restricted to carrier phase based positioning.RTKLIB  is used The RMS values of errors in 3D positioning are 117 mm and 0.7 mm for uncompensated and compensated beamformers respectively after convergence, which shows a significant improvement in the compensated beamformer.However, this level of accuracy is achieved by neglecting the error contributions from carrier phase multipath, non-ideal antennas and other residual differential errors. In order to assess this effect on accuracy, another test is performed by adding carrier phase multipath and antenna array imperfections into the simulated data. Residual errors due to non-ideal antenna elements, coupling and other imperfections after estimating the compensation matrix are modeled based on the analysis in Since the proposed method does not bring in any phase distortion due to the spatial processing to any of the incident signals including reflected signals, the carrier phase multipath error model stays unaltered. Therefore, in order to simulate close to actual signal conditions, a modeled carrier phase multipath is added to the error-free carrier phase measurements. The procedure adopted to obtain the carrier phase multipath model is as follows. A differential static GPS survey was conducted over a distance of 20 km. Two NovAtel Propak-V3 receivers and geodetic antennas were used to collect data 20 times per second. The carrier phase ambiguities were resolved as integer values and a least-squares method (to avoid smoothing effects) was used to generate the double difference carrier phase residuals, which include noise, differential atmospheric effects and multipath. High frequency thermal noise is at the sub-millimetre level but was nevertheless removed using a low pass filter. The differential atmospheric effect is expected to be at the millimetre level at the most. The remaining residuals constitute a valid measure of carrier phase multipath and are shown in These multipath effects and antenna array imperfections were added to the phase measurements obtained in the receiver after beamforming. The proposed beamformer functions are independent of the architecture of the receiver being used. Therefore, any multipath mitigation method can be added to the output of the proposed beamformer. Additionally, when the receiver is equipped with an antenna array, spatial domain processing methods could be used to alleviate these errors, which have not been investigated in this paper.This paper investigated the phase distortions in adaptive blind beamforming. The results showed that, although the conventional blind beamformer can effectively suppress jammers or interference, it is not immune to carrier phase distortion.Herein, a solution for the above problem was proposed. The proposed approach took advantage of the symmetry in the chosen array configuration, phase compensation, and conjugate symmetric array gain vector constrained optimization. Contrary to similar work, it neither requires the array to be fully calibrated nor imposes any modifications on a GNSS receiver architecture.The experiment, which was performed using the sample data generated from the combination of hardware and software simulators, showed that the new method resulted in position estimates at millimetre-level accuracy for the blind beamforming method in the interference scenario and at the few millimetres level when multipath and non-ideal antennas are considered. Moreover, errors on the position solution were compared between the new approach and the conventional blind beamformer using a multi-antenna software receiver developed during the course of this research. For comprehensive evaluation, Monte Carlo simulations were performed to analyze performance of the proposed method in different conditions.


Process-based models are effective tools to predict the phenotype of an individual in different growing conditions. Combined with a quantitative trait locus (QTL) mapping approach, it is then possible to predict the behavior of individuals with any combinations of alleles. However the number of simulations to explore the realm of possibilities may become infinite. Therefore, the use of an efficient optimization algorithm to intelligently explore the search space becomes imperative. The optimization algorithm has to solve a multi-objective problem, since the phenotypes of interest are usually a complex of traits, to identify the individuals with best tradeoffs between those traits. In this study we proposed to unroll such a combined approach in the case of peach fruit quality described through three targeted traits, using a process-based model with seven parameters controlled by QTL. We compared a current approach based on the optimization of the values of the parameters with a more evolved way to proceed which consists in the direct optimization of the alleles controlling the parameters. The optimization algorithm has been adapted to deal with both continuous and combinatorial problems. We compared the spaces of parameters obtained with different tactics and the phenotype of the individuals resulting from random simulations and optimization in these spaces. The use of a genetic model enabled the restriction of the dimension of the parameter space toward more feasible combinations of parameter values, reproducing relationships between parameters as observed in a real progeny. The results of this study demonstrated the potential of such an approach to refine the solutions toward more realistic ideotypes. Perspectives of improvement are discussed. Complex phenotypes of plants are not only regulated by both multiple interacting genes and environmental conditions but also by a series of interactions, competitions and feedbacks operating at different levels within the plant. In this context, modelers have the ambitions to synthesize information coming from complex genomics data sets, physiology and biochemistry with environment responses into mathematical models. Moreover, modeling can help to weave their way through this complexity and predict phenotypic consequences of changes at different levels . The recTo meet this challenge, several difficulties must be overcome. The first one consists in successfully integrating the genetic control in process-based plant models. Various research groups have worked on this integration for many years see .Next step concerns the identification of individuals displaying best performance in given conditions. Depending on the crop, performance may be described as a set of targeted traits that may prove to be antagonistic. In this case it is difficult to forecast which traits or processes to favor in order to get the best tradeoff. Using process-based models but lacking a dedicated method, one may run 1000s of simulations, combining values of parameters in all directions, without ever approaching optima. Fortunately, optimization algorithms are efficient tools to solve such problems. In a pioneering work, Nature-inspired optimization algorithms  are increasingly used . These aHowever, a major locking point persists on the extension of the second step to better consider the complex genetic architecture controlling the model parameters into account in the optimization scheme. Indeed, in a ‘Gene-to-Phenotype’ modeling approach, the simulations performed by One way is to include known genetic constraints in the definition of the space of parameter variation that is explored during the optimization step. This option is being explored by The present paper constitutes a first step toward the complete scheme of integration of process-based and genetics models associated with an optimization algorithm to design realistic ideotypes. A multi-objective optimization algorithm was used to find best allelic combinations to enhance three traits simulated by a peach fruit quality model. In a framework of a biparental progeny , seven parameters of the process-based model were estimated for each of the 159 individuals and QTL detected for all of them. A genetic model including the allele effect of the wild parent at each locus for each parameter, pleiotropic effects of some loci and linkage between loci was developed. For sake of simplicity, epistasis was not considered in this study. The genetic model was combined with the ‘Virtual Fruit’ process-based model and coupled with the NSGA-II algorithm to solve the continuous  and the combinatorial  optimization problems respectively. We compared the spaces of parameters obtained with different tactics and the phenotype of the individuals resulting from random simulations and optimization in these spaces. The features of the virtual phenotypes selected from the optimization steps were compared between the different tactics and discussed in sake of feasibility.Supplementary Table S1).The process-based simulation model used is the ‘Virtual Fruit’ model which integrates many sub-models dealing with fruit growth and quality elaboration . In thisSLA, m2g-1). The latter is considered to be independent of tree growth and environment but under genetic control. The daily available pool of C assimilates consists of leaf assimilation plus possible C mobilized from reserves. The leaf photosynthesis rate may be affected by feedback inhibition through the leaf storage reserves. Fruit carbon assimilation and respiration are also considered. If required, reserves are mobilized first from the leafy shoot, then from the 1-year-old stem. Carbon is allocated according to organ demands and priority rules.The leaf area is computed as the product of dry weight of the structural part of leaves and specific leaf area relative to the structural part of leaves .DM) growth is modeled in terms of growing degree-days (GDD) using the following logistic equation:The potential fruit dry matter (where:fruitDM: is fruit dry mass (g) at date jd,A and B: are parameters involved in the calculation of the logistic asymptote fixing final dry mass of fruit; their units are respectively (g) and (dimensionless),ini: is the initial relative fruit growth rate (GDD-1),RGR3: is the date of the inflection point of the curve of fruit growth (GDD),Pj: is the date (GDD),d0: is the initial dry mass of fruit (g),DMj0: is the initial date (GDD) corresponding to DM0.dThe values of these parameters are used to calculate the fruit carbon demand for each genotype.stoneDM) is computed as:The stone dry mass (stone is a parameter (dimensionless) and Wstone (g) is a genotypic parameter which corresponds to potential maximal stone mass (plateau of the curve).where kThe stone ratio is then computed using (1) and (2).sugar) is computed from the carbon used for dry flesh growth rate and the carbon used for synthesis of other carbohydrate compounds .The variation in total amount of carbon (g) in the fruit flesh as sugars (CCsupp is the flesh carbon supply (gC day-1) and ksugar (day-1) is the relative rate of consumption of carbon as sugars in the fruit flesh for synthesis of compounds other than sugars. The parameters α (gC gDM-1) and β (gC gDM-1) are the respiration coefficient of fruit growth and the carbon concentration in fruit biomass, respectively.where FFM)-1), is computed as:The total sugar concentration, SU (g (100 gTS is an average carbon content of sugars (gC/g sugars) for sucrose, glucose, fructose, and sorbitol. FMflesh is the flesh fresh mass (g).where σThe changes in flesh fresh mass are predicted as a function of turgor pressure by the Lockhart equation. The turgor results from the balance of water flows. The flow of water to the fruit is simulated considering full water availability for the plant. This flow is driven by differences of hydrostatic and osmotic pressures between stem and fruit and the fruit volume changes are predicted as a function of turgor pressure by the Lockhart equation. The flow of water leaving the fruit (transpiration) is calculated from skin conductance to water vapor.The outputs used in this work are three fruit traits: dry mass of fruit (g), the part of stone in the fruit (in dry mass) and the total sugar concentration of the flesh. The model is driven by its parameters, which are constant over time and supposed independent of the environment. The simulations were performed for the year 2009 from 87 days after bloom (DAB) to 150 DAB. The parameter values were obtained from the literature or estimated for peach based on previous studies .DM growth, i.e., A,B,RGRini,P3, two involved in the computation of the ratio between stone and fruit dry mass, i.e., kstone,Wstone and one involved in the calculation of carbon assimilation from sources, i.e., SLA . The choice of the first four parameters has been done based on deep sensitivity analysis of the used model. This choice was made based on two criteria: the main effect of each parameter on model outputs and the interaction terms with other parameters. We used different sensitivity methods and according to their results, we selected the parameters which show significant effects on the outputs of the model. The three remaining parameters were selected as being involved in some important biological processes.We selected seven parameters: four involved in the Prunus davidiana P1908 (D) and Prunus persica cv. ‘Summergrand’ (S) and then cv. ‘Zéphyr’ (Z) as described by The peach genotypes studied come from a population obtained by two subsequent back crosses between fleshDM) was determined after drying flesh for 72 h at 70°C to constant weight. To compute dry fruit, flesh and stone masses for each monitored fruit, several allometric relationships between fruit diameter and fruit dry mass, stone dry mass and fruit dry mass, were used for each genotype , using the ‘lnme’ R package which allows fitting non-linear mixed-effects models by maximum likelihood. This way we obtained a matrix including the values of the seven parameters for each of the 159 individuals of the progeny.Using the process-based model described above and this matrix, we could simulate the kinetics of dry mass growth of fruit, stone ratio and total sugar concentration.DM), minimize the stone ratio (SR), and maximize the sugar concentration (SU). This multiobjective optimization problem was formulated as follows:The model was used to find the set of the seven parameters allowing the optimization of the three traits of interest: maximize fruit dry matter mass T is the vector of the parameters and D is the domain of variation of these parameters defined by lower and upper bounds. The negative sign before DM and SU objectives were introduced to ensure maximization of these objectives in a formulation based on minimization.where X = . All of the QTL detected for a parameter accounted for between 10 and 50% of the observed variation depending of the parameter (Supplementary Table S2). For each parameter, the effect of the presence of the D or S allele at each genetic locus linked to each QTL detected for the parameter was computed using a linear regression using ‘lm’ function in R. The resulting genetic model includes 31 different loci, since five loci are involved in two or three QTL of different parameters. This allows taking into account pleiotropic effect of some loci with contrasting effects on the different parameters. Those loci are positioned on the eight linkage groups and they are more or less distant from each other.For sake of proof of concept in this study, we kept all the QTL displaying a LOD score >1 (but only 1 per linkage group), so as to build a genetic model with a consequent number of loci. Thus a total of 37 QTL were kept  at each of the 31 loci included in the genetic model, it is possible to calculate the corresponding values of the seven parameters of the process-based model. Thus, the genetic model was integrated in the ‘Virtual Fruit’ model. The values of the seven parameters are calculated from the alleles present at the QTL using the genetic model consisting of the following linear system of equations:loc[x], with × from 1 to 31, take the value 0 or 1 depending on the presence of D or S allele at the QTL, respectively.where the coefficients correspond to the additive effect of each QTL, and The probability to dissociate alleles of neighboring genes by obtaining genetic recombinants decreases gradually as the physical distance between the genes declines. The detailed analysis of the QTL results together with the genetic map dataset revealed the implication in the genetic model of loci closely linked. To take into account these supplementary constraints in the optimization step, we considered that distant loci less than 12.5 cM were inseparable. This led to the addition of 14 constraints and resulted in the definition of haplotypes including series of loci.DM mass, minimize the stone ratio, and maximize the sugar concentration. This multiobjective optimization problem was formulated as follows:The enriched model  was used to find the best binary combinations of the 31 loci optimizing the three traits of interest: maximize fruit loc is the vector of 31 loci, ε is a very small real, i.e., 1e-9 number and insloc is the set of 14 inseparable loci pairs given in the Table 1. The last formula represents the fact that for each of the 14 loci pairs, the two loci must take the same value (0 or 1) since they are considered inseparable and inherited from the same parent.where We used NSGA-II algorithm to deal with both optimization problems formulated above. We used NSGA-II as it is considered in the optimization community as the reference thanks to its performances. Also, NSGA-II allowed us to tackle both continuous and combinatorial problems using the same algorithm while others MOEAs were developed especially for only one type. The NSGA-II algorithm works by randomly creating a parent population P0 and sorting it based on non-domination. To sort a population of N individuals according to the level of non-domination, each solution is compared with every other solution in the population. All individuals in the first non-dominated front are thus identified (F1). Then, the solutions of the first front are temporarily discounted, and the above procedure is performed again to find individuals for the next front (F2). The procedure is repeated for subsequent fronts  until all individuals are assigned to their ranks. The fitness is set to a level number: the lower the level the higher the fitness (F1 is the best). Next, the NSGA-II uses binary tournament selection, recombination, and mutation operators to create a child population Q0 of size N. At each subsequent generation, t, the algorithm merges the parent Pt and the child Qt populations into a combined population Rt of 2N individuals and sorts Rt according to non-domination, as described above. To create the next parent population, the NSGA-II uses the crowded comparison operator to select only N solutions from the Rt population. The new population Pt+1 is subsequently used for selection, crossover and mutation to create a new population Qt+1 of size N. The above procedure is continued until a predefined number of generations are created .2) where M is the number of objectives involved in the optimization problem to be solved and N is the size of the population. The interested readers can refer to the above cited references which give more details on NSGA-II. Parameters’ setting has been achieved based on the suggestions of the original version of NSGA-II for some parameters and on authors’ previous works for others. The NSGA-II parameters used in this paper are presented in Table 2.The complexity of NSGA-II is O whose boundaries were fixed to the observed minimal and maximal values observed in the progeny. This parameter space was explored by random draw  and by an optimization procedure.In a second step, a restricted parameter space (‘parameters_restricted’) was defined by fixing its boundaries to the minimal and maximal values made possible by the genetic model. Selecting all the alleles decreasing (increasing) the values of one parameter, we extracted the minimal  boundary of this parameter and defined a space representing the larger subset of values that one may hope rebuild from allele combinations. This space was also explored by random draw and optimization.In a third step, on the basis of the genetic model we defined an allele space consisting in a matrix of 31 loci taking the values of either 0 or 1. This ‘alleles’ space was explored by random draw and optimization.In the fourth and last step, linkage between loci was taken into account considering that loci closer than 12.5 cM could not be dissociated. This resulted in 14 constraints to include in the optimization procedure .Table 3 and the boundaries of the different spaces in Table 4. The phenotype of the individuals resulting from random simulations and optimization in these spaces were compared.The number of simulations and solutions obtained by the optimization algorithm are presented in The quality fruit model was used in this study to simulate the three targeted traits: fruit dry mass, stone ratio and total sugar concentration.stonek and A . The set of the seven parameters enabled to simulate the large variability of phenotypes observed in the progeny for fruit dry mass, stone ratio, and total sugar concentration.On the basis of phenotype measurements, seven parameters of the process-based model were parameterized for the 159 individuals of the progeny. The parameter values and simulated traits were compiled in the ‘parameters_progeny-fits’ dataset. The distributions of the parameter values were sometimes uneven, especially for Figure 1 _ gray dataset).Within the ‘parameters_obs-bounds’ space, 500 sets of random values were drawn for the seven parameters and used to simulate the three targeted traits , part of the individuals from the ‘parameters_progeny-fits’ dataset was not reproduced by the ‘parameters_random_obs-bounds’ dataset. Hence, the individuals with relative large stone were not obtained by random drawing, suggesting that other parameters of the process-based model involved in stone elaboration may be genotype dependent.When looking to the stone ratio .The ‘parameters_obs-bounds’ parameter space was also explored thanks to the optimization procedure in order to find solutions which enhanced the three targeted traits (‘parameters_optim_obs-bounds’ dataset): 1193 distinct solutions were found. Four of the seven parameters had values nearly fixed toward one boundary of the parameter space, although the three other ones were distributed in a more or less large range of the space  but all better than the individuals obtained by random drawing of parameters. This emphasized the benefit of the optimization step to design combinations of parameters resulting in good phenotypes. It would require a very high density of random points to get individuals in this area. Indeed, best phenotypes stood in a zone hardly explored by random exploration suggesting highly refined combinations of parameters . The very large number of different parameter combinations compared to the very low variability of the resulting phenotypes pointed out the tight relationship between the parameters A and B , and the low influence of the parameter SLA on the targeted traits. In contrast the parameters stoneW, stonek, iniRGR, and 3P appeared as prevalent in the determination of the targeted traits.The optimization procedure allowed the detection of 1193 distinct individuals very similar in terms of phenotypes .The best individuals of the random dataset and even more the optimized solutions were better than the observed individuals of the progeny. Indeed in this approach not one constraint was taken into account, neither physiological nor genetic, probably leading to unrealistic combinations of parameter values and outrageous phenotypes located in an unrealistic zone . The optimization procedure ended up again on a large number of very similar individuals. Comparing the 1370 optimized individuals obtained  to the corresponding random landscape, similar conclusions to those stated previously on the strength of the optimization procedure can be drawn. Compared to the solutions obtained previously (‘parameters_optim_restricted’ versus ‘parameters_optim_obs-bounds’), the solutions were closer to the space explored by the “parameters_progeny-fits” dataset . Thus, the benefit procured by this step is the elimination of extreme individuals that may never be obtained.The random individuals from this parameter space were very similar to the individuals observed in the progeny for the three targeted traits . Hence the parameter B, controlled by only two loci in our genetic model may take four different values, whereas the parameter stonek controlled by seven loci displayed a quasi-continuous distribution. When regarding the distributions of the three traits, the random individuals were very diverse, almost as much as the ones from the ‘parameters_random_restricted’ dataset . Hence, the switch from a ‘parameter space’ to an ‘allele space’ did not reduce drastically the range of phenotypes.Random drawing of allele combinations in this space allowed us to build the ‘alleles_random’ dataset. The more loci involved in the definition of a parameter, the more the resulting distribution of the values of this parameter was quantitative (31). In contrast with the two previous datasets obtained from the optimization of the values of the parameters, the resulting distinct individuals were few, only 170 . The latitude of possible solutions was very narrow; however, those individuals outranged again the corresponding random landscape for the three targeted traits .The optimization procedure explored in record time this allele space of more than 2 billion solutions . Compared to observed individuals from the ‘parameters_progeny-fits’ dataset (green), these optimized individuals reached another compromise of the three targeted traits: they displayed lower values than best observed values for dry fruit mass and total sugar concentration but part-cons lower values of stone ratio.The resulting solutions were very few, only 14 , and, as expected, they were worse than the ones from the ‘alleles_optim’ dataset, especially for the stone ratio  into a very shaped landscape  which draws the space of possible solutions with a physiological point of view, constraining traits with compromises. Although, the individuals from the three random datasets looked nested with the ‘parameters_progeny-fits’ dataset in terms of values of parameters, they proved to be quite different when looking to the simulated traits resulting from the model. But they all aligned along borders that look impenetrable and which, without any doubt, mark physical, and/or physiological limits driving the fruit system.As an illustration, the peach fruit quality model transformed clouds of parameter values  reproduced quite well the tendency observed in the real progeny (green dataset), illustrating the strength of the genetic model . As for the relationship between A and B, the number of QTL detected for B did not allow reproducing properly the relationship. Hence, the switch from the ‘parameters_restricted’ space to the ‘allele space’ possibly added interesting effects on the reproduction of biological constraints conducting to the construction of more realistic phenotypes. Regarding data from the ‘parameters_optim_obs-bounds’ dataset (black dataset), they appeared out of the bounds of the observed relationships. In particular, due to the remoteness from the observed space for iniRGR and 3P, and due to the shift of the relationship between A and B, the optimized individuals seemed unconceivable.The genetic model allowed us to reproduce somehow the genetic links between the parameters that were observed in the progeny, which resulted in a reduction of the realm of possibility. Indeed, the relationship between A and B on one hand and iniRGR and 3P on another hand. In addition, our results showed sometimes clear monotonic relationships between one trait to optimize and a parameter. In the absence of antagonist effect of the parameter on another objective, the optimization step rationally resulted in parameter values heading to the boundaries. However, we chose to keep working with the seven parameters (instead of proceeding to parameter reduction) as we were investigating the genetic architecture and effect of the QTL controlling these parameters. The QTL controlling dependent parameters were indeed not completely similar and reducing the model at this stage, as done by As previously mentioned, four parameters have been selected based on their main effects and interaction terms using sensitivity analysis. Three parameters have been selected as being involved in some important processes from agronomic and genetic points of view. Doing so, dependencies among parameters were expected. We actually identified correlations between Taking into account the genetic model allowed restraining the space of solutions from an unrealistic landscape (gray dataset) to a more feasible one , and thus designing optimized solutions that are more likely to be obtained .Similar results were obtained by One of the main difficulty to progress in this sense persists to be the estimation of the parameter values for 100s of individuals despite the elaboration of phenotyping platforms . Indeed,The optimization procedures used in this study proved to be very efficient to converge in the zones of the spaces explored that resulted in phenotypes outranging the corresponding random landscape, located at the margin of what could be impenetrable frontiers.Finally, the solutions in terms of parameter values that appear most probable  fell into a space which was explored both by random individuals  and real observed individuals. This is quite promising for the chance to be able to obtain them by crossing.The consideration of a bi-allelic progeny as the basis of the development of the genetic model is a simple case that dramatically reduces the genetic diversity taken into consideration. Especially in this case-study, only the effect of the presence or not of the wild allele was estimated. The case of an F2 population, for example, would at least add a little bit more of complexity in the genetic model, with three types of genotypes (two homozygotes and one heterozygote) and the issue of dominance versus additivity effects to consider. The latter issue was taken into account in Finally, the study proposed here is imperfect since it did not take into account the probabilistic character of the linkage between loci. Thus, a further step to progress toward assisting plant breeding would be the development of an optimization procedure based on probabilistic rules that would allow building allele combinations according to distances between loci and thus rank the solutions according to the chance to obtain them. For this purpose and to develop a full breeding optimization system, two main ideas will be investigated in our future work. The first one will consist in integrating the stochastic (probabilities) constraints on loci linkage into the genetic model itself. Thus the values of the parameters predicted by the genetic model would take into account the linkage constraints. This is the option followed in quantitative genetics simulation systems as developed by The second one will be to consider those constraints in the optimization step  and to deal with the new stochastic optimization problem under constraints of equality of alleles between near loci. The resulting problem will be harder to solve and will need specific tools  and techniques (constraints relaxation for example) to deal with.Developing an efficient tool to predict GxE environment and design ideotypes adapted to particular environmental conditions that the breeder has good chance to succeed to create is one important challenge for the coming years. The use of a process-based model combined with a genetic model and associated with an efficient optimization procedure is a promising path provided that the two models are robust and accurate.BQ-T, MG, and M-MM conceived and designed the work; PV and M-MM devised the algorithms and performed the simulations; BQ-T analyzed the data; BQ-T, MG, and M-MM interpreted data; BQ-T, M-MM, and PV wrote the paper; MG revised it critically for important intellectual content; BQ-T, MG, PV, and M-MM approved the final version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


A deep characterization of neurological patients is a crucial step for a detailed knowledge of the pathology and maximal exploitation and customization of the rehabilitation therapy. The muscle synergies analysis was designed to investigate how muscles coactivate and how their eliciting commands change in time during movement production. Few studies investigated the value of muscle synergies for the characterization of neurological patients before rehabilitation therapies. In this article, the synergy analysis was used to characterize a group of chronic poststroke hemiplegic patients.Twenty-two poststroke patients performed a session composed of a sequence of 3D reaching movements. They were assessed through an instrumental assessment, by recording kinematics and electromyography to extract muscle synergies and their activation commands. Patients’ motor synergies were grouped by the means of cluster analysis. Consistency and characterization of each cluster was assessed and clinically profiled by comparison with standard motor assessments.Motor synergies were successfully extracted on all 22 patients. Five basic clusters were identified as a trade-off between clustering precision and synthesis power, representing: healthy-like activations, two shoulder compensatory strategies, two elbow predominance patterns. Each cluster was provided with a deep characterization and correlation with clinical scales, range of motion, and smoothness.The clustering of muscle synergies enabled a pretherapy characterization of patients. Such technique may affect several aspects of the therapy: prediction of outcomes, evaluation of the treatments, customization of doses, and therapies. Neurological diseases are one of the main sources of disability, especially in Western countries. A wide variety of pathologies and symptoms can lead to partial or complete disability, which influences people life up to preventing self-autonomy. A recent state-of-the-art review on rehabilitation  has been widely studied on stroke subjects and many alterations have been found compared to healthy individuals. EMG is an interesting assessment since, despite its limitations in terms of cross-talking, electrode placement, repeatability , factor analysis (FA), independent component analysis (ICA), and non-negative matrix factorization (NMF)  account for the majority of the EMG activity related to specific tasks. Previous studies investigated specific aspects of functional movements, with deep analysis on frontal reaching , Italy, during the period ranging from years 2014 to 2016. The study was reviewed and approved by the local Ethics Committee at Lecco Manzoni Hospital and was conducted in compliance with the Declaration of Helsinki. Written informed consent was obtained from each subject before inclusion in the study.Twenty-two poststroke patients in the chronic phase (more than six months from the stroke event) were recruited for the study. All patients had motor deficits in one upper limb. No requirement on motor functionality was requested, even if patients were selected, with different levels of impairment. Eligibility criteria also included comprehension of the tasks to be performed. The demographic characteristic of the patients are listed in Table A Flowchart illustrating the outline of the study is shown in Figure BTS Smart-D system for kinematics (recorded at 140 Hz) and EMG ;The following equipment was used in this study:Kinematics was recorded for D5 and C7 vertebras, acromion, elbow epicondyle, styloid process of the ulna.s-EMG was recorded on the following Selected Muscles: upper trapezius (Tr), pectoralis major (Pm), deltoid anterior (Da), deltoid middle (Dm), deltoid posterior, triceps brachii caput medialis, biceps brachii caput longus, and brachioradialis (Br).Clinical evaluations were performed by a physical therapist using the FMA , and the second one representing time-variant activation commands for each synergy (ic) . Data from eight s-EMG channels were recorded. Data were filtered with a Hilbert transform filter that allows the loss of the minimal amount of signal. EMG data from each subject and each trial were pooled together in a single aggregated matrix and synergies were extracted using the NMF algorithm  represents the EMG data at time t and N is the total number of extracted synergies.where for each of the recorded muscles, EMG. For each r, the NMF algorithm was applied 1,000 times in order to avoid local minima and the repetition accounting for the higher variance of the signal was chosen as the representative of order r. The number of synergies was chosen as the minimum r explaining at least 0.80 of the variance of the signal  for at least two reasons. First, many studies  or because it fractionates into two or more synergies. In such cases, the choice of S1 was qualitatively driven by the following criteria: (1) entity of the activation (preferred on severely impaired patients who have limited ROM) and (2) timing of elicitation of the synergy (preferred for patients who present fractionation issues). To authors’ opinion, while still being improvable, this procedure allows a coherent method for the profiling of the clinical status of patients in the framework of muscle synergies.k-means Matlab cluster algorithm. The algorithm was applied to an aggregated matrix containing the flexion synergy of each patient, selected according to the criteria discussed in the previous paragraph. Each possible clustering order, ranging from 2 (minimum) to 22 (maximum), was tested, by repeating 200 times the algorithm and selecting, for each order, the solution with higher Silhouette Score Parsimonious number of clusters for synthesis power .(2)Silhouette Score, indicating the goodness of the clustering, as a synthetic index for each clustering order. The higher the Silhouette, the better elements fit to their cluster.(3)Mean scalar product among all the clusters for each cluster order. The lowest the mean dot product, the more cluster are differentiated. Too similar clusters might be grouped together decreasing the order of the clustering.(4)If possible, single-patient cluster solutions are avoided or at least limited.Then, the selection of the appropriate number of clusters was made by pondering the following metrics:Being metrics (1), (2), and (3) not directly comparable, authors decided to select the lowest number of clusters according to whom both the Silhouette Score and the Mean Scalar Product would increase and decrease, respectively. If both conditions are met, increasing the number of clusters has certainly lead to better clustering. Otherwise, increasing the number of clusters may not be needed.Lastly, extracted clusters are compared to FMA and RPS assessment for detecting correlations between the EMG-based analysis and standard assessments for discussion and further characterization of patients. Furthermore, a comparison with basic ROM assessment was performed. Shoulder flexion angle in the sagittal plane and the elbow extension angle, both measured at maximum shoulder flexion—end of the reaching movement  proposed in the paragraph 2.7, a maximum of 11 clusters was considered, corresponding to 1/2 of the sample. This way, there is the possibility that every centroid is populated by at least two patients. Increasing further the number of clusters would prevent any kind of generalization, and consistently, were Clustering Composition for clustering order = 5 is reported in Figure Table Detailed FMA scores for each of the five clusters is reported in Figure Detailed RPS scores for each of the five clusters is reported in Figure Table Detailed Normalized Jerk scores for each of the five clusters is reported in Figure In this article, muscle synergies were extracted and then clustered from a cohort of chronic stroke patients to identify motor patterns for the execution of 3D reaching movements. The muscle synergies method was selected to understand how to characterize the pathological movement and link it, if possible, to the clinical status expressed by clinical scales and kinematics. Compared to less sophisticated EMG analysis, muscle synergies method provides more information about the relationship existing between the muscles, either for what concerns the amplitude and cocontractions, either for the timing of activation that can lead to different interpretation about the role of the muscle during the movement execution . All these features are considered in coupling by synergies extraction, adding valuable analysis in respect to single EMG stream analysis. Furthermore, the synergies analysis provides an analytical framework which is coherent with the theories of motor learning and production related to the modular organization of the brain Schmidt, , and to As for other matrix factorization methods reported in the literature , which is replaced by the Tr. Such a pattern, found in chronic patients, might be addressed to the capability of compensating the lack of Da activation to perform the movement. Typical effect of such group is scapular elevation. Cluster 2 groups moderate impaired patients that rely on compensatory strategies. With the exception of Patient 16, this cluster appears to be quite homogeneous both with respect to the RPS and the FMA. The majority of the patients belonging to Cluster 2 have full or nearly full ROM at the shoulder, even if two of them do not complete the elevation of the arm. All of them do not extend completely the elbow. The mean Normalized Jerk within the cluster is low, indicating a quite smooth movement execution, probably to due simplified control related to the uncontrolled, fast elevation relying mainly on Tr. Pt15, suffering of dystonia, represents an expected exception. Cluster 2 appears to have high similarity to Cluster 1 (as reported in Table Cluster 3 shows patients who strongly compensate by activating the Pm muscle. Their ROM is often not complete (with the exception of Patient 4) and the compensation may lead to shoulder adduction and intrarotation. Contrarily to previous clusters, the functional level presents high variability but, overall, is consistently lower than the one of previous clusters, as shown by clinical scales. Prognostic indications coming from the belonging to this cluster might be worse if compared to Clusters 1 and 2; main elevator agonists in the sagittal plane play less relevant role than the pectoralis that is not an elevator. However, it should be noted that the composition of this cluster is not homogeneous in terms of clinical scales. Such finding is confirmed by kinematic parameters that range from very good ROM and smoothness (Pt14) to very low performances (Pt20). Cluster 3 is probably the less uniform one.Cluster 4 is a single-patient cluster having low proximal functionality, characterized by the prevalence of elbow/distal muscles in the elevation phase. The patient present very low RPS and FMA scores, and have very limited/null ROM in respect to rest/equilibrium poses. The patient belonging to this cluster shows prevalence of triceps activity and lacks muscular tone needed to perform the movement. He is likely to show reduced recovery; kinematics indexes indicate very low ROM and smoothness.Cluster 5 is a two-patient cluster. Patients belonging to Cluster 5 show low proximal functionality, characterized by the prevalence of elbow/distal muscles in the elevation phase. Such patient present extremely low RPS and FMA scores and have very limited/null ROM in respect to rest/equilibrium poses. Patients belonging to this cluster show prevalence of elbow flexors muscles (biceps and Br) activity and lacks muscular tone needed to perform the movement. They are likely to show poor prognostic outcomes, since kinematics shows extremely reduced ROM and low smoothness.It might be claimed that the sample of patients in this study splits into two main groups: patients with shoulder prevalence and elbow prevalence. Each group further splits depending on the compensatory strategy used for shoulder flexion  and elbow prevalence (flexors and extensors). This data clustering might be a valuable starting point for interpreting patients’ performance and motor recovery.via the recruitment of additional agonist muscles, thereby distributing the muscle force on other agonists  reaching movements performed by stroke survivors .As a global remark, and main result of the study, authors observed that muscular synergies profiling does not match precisely the evaluation provided by clinical scales, suggesting that the evaluation provided by standard tools should be integrated for complete assessment and patients characterization. Consequently, clinical scales might be insufficient for correct and deep patients’ profiling and therapy customization.Authors observed that, starting from quite low-order clusters (5) algorithms for clustering extraction tend to create single-patient centroids (at least in the group of patients examined in this study). Order 7 clustering individuates three single-patient-based centroids. Such findings denote the difficultness in grouping patients that are in general characterized by their own peculiar muscular patterns. Consequently, cluster analysis suggests that patients tend to show individual patterns. However, order 5 clustering was considered as a reasonable grouping order to provide deep enough characterization. It detects a centroid characterized by healthy-like activations, two major compensatory strategies, and two severely impaired groups.Few studies in the literature investigated the usefulness of a detailed knowledge of patients’ motor capabilities for a pretherapy detailed assessment. Such a feature would give the clinician the capability of selecting a sequence of interventions that proved in the past to be effective on that specific group of patients. Muscular synergies have relevant potential under this point of view. They are extracted directly from the stimulation given by the nerves to the muscles on the patient and are not the result of tests, clinical scales or kinematic outcome variables. Furthermore, synergies approach can be tuned by selecting, either (1) synthetic approaches, trying to reduce the number of clusters and classifications to synthetize groups and interventions ; (2) analytic approaches, detailing many more clusters for capturing even minor motor differences . Interestingly, the synthetic approach could be useful to identify macrodivisions, such as deciding if a patient is more suitable for a specific therapy approach in a restricted range of choices. On the contrary, the analytic approach might be chosen to specify the more suitable detail on refined customization.To authors’ knowledge, no studies, either in acute or in chronic stage stroke patients, have explored the possibility to characterize the patient and the therapy merging two or more methods. Especially in the upper limb rehabilitation field, where less results are usually obtained in comparison to the lower limb, an integrated approach which combines more variables and different levels of evaluation, from anatomical to functional, could be the solution for a real therapy customization according to the predicted outcomes. Muscle synergy characterization could play a valuable role in prediction of prognostic outcomes, also in association with previous techniques. In frontal reaching movements, patients with prevalence of shoulder synergies, dominated by elevator muscles , might be candidates for motor recovery. Prevalence of elbow activity is instead a bad prognostic index, indicating the impossibility to perform the gesture. Such patients would need antigravitary support to perform reaching movements. For the reasons explained, synergies could be integrated in a multifactorial assessment to better estimate the residual potential of the patient and consequently suggest the best rehabilitation path available. Moreover, synergies extraction could be useful for a real-time EMG-based robot interaction for an improved compliance during a robotic rehabilitation protocol. Potential application includes pathology-specific synergy characterization can be useful also for an efficient EMG-based robotic control: the real-time patient’s EMGs could be mapped in the synergy space, with real time estimation of the time-varying activation coefficients  or Further studies will test the method proposed in this article for the assessment of the effect of robotic therapy approach to upper-limb rehabilitation. Modifications of clusters and shifting from one clusters to another will be considered as valuable insights for assessing the efficacy of the therapy.This study was carried out in accordance with the recommendations of local Ethics Committee at Lecco Manzoni Hospital with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the local Ethics Committee at Lecco Manzoni Hospital.AS designed the experiment, wrote the software for synergies extraction, performed the experimental campaign, elaborated the data, and wrote the article. AC performed the experimental campaign and participated to data analysis and interpretation. MM was responsible for the research project that funded the work. He revised the article and participated to data analysis and discussion. LT was the head of the CNR research group. He revised the article and participated to data analysis and discussion. FM is the head of the Villa Beretta Hospital and participated to the conception of the evaluation system, and gave clinical interpretation to the data.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.