Peripersonal space (PPS) representation is modulated by information coming from the body. In paraplegic individuals, whose lower limb sensory-motor functions are impaired or completely lost, the representation of PPS around the feet is reduced. However, passive motion can have short-term restorative effects. What remains unclear is the mechanisms underlying this recovery, in particular with regard to the contribution of visual and motor feedback and of interoception. Using virtual reality technology, we dissociated the motor and visual feedback during passive motion in paraplegics with complete and incomplete lesions and in healthy controls. The results show that in the case of paraplegics, the presence of motor feedback was necessary for the recovery of PPS representation, both when the motor feedback was congruent and when it was incongruent with the visual feedback. In contrast, visuo-motor incongruence led to an inhibition of PPS representation in the control group. There were no differences in sympathetic responses between the three groups. Nevertheless, in individuals with incomplete lesions, greater interoceptive sensitivity was associated with a better representation of PPS around the feet in the visuo-motor incongruent conditions. These results shed new light on the modulation of PPS representation, and demonstrate the importance of residual motor feedback and its integration with other bodily information in maintaining space representation. Acquired spinal cord injuries (SCIs) typically cause varying degrees of impairment in sensory and motor functions in the body parts below the lesion, potentially also involving complete paralysis and anaesthesia.3. However, if research is limited to studies on these patients, it means that it is not possible to distinguish between the effects relating to brain damage and those relating to motor deefferentation and somatosensory deafferentation. Thus, in order to shed light on the role of sensorimotor information in cognition, it is crucial to test cognitive modifications in patients with sensorimotor impairments but without brain damage, a condition that is typical of SCI patients. An increasing number of reports have focused on the effects of SCI on body- and action-related cognitive representations, including changes in action representation6, motor learning8, motor imagery10 and body representation18. Interestingly, these studies indicate that SCI induces changes that, in terms of cognitive representation, may lead to both an increase in the functions of body parts above the neurological lesion level  and a decrease in functions below the neurological lesion level6.To date, the impact of sensorimotor disorders on cognitive processing has been for the most part neglected or only tested in patients with brain lesions20 p. 139) is reduced around the body parts below the lesion level (e.g. around the feet), but preserved around the body parts above the lesion level (e.g. the hands)21. Even if only few studies have focused on PPS around the feet, these indicate its presence in healthy subjects24.For example, in cases of SCI, Peripersonal Space (PPS)  or against the thumb (lower elevation). Distractor lights were fitted onto a wooden frame for the feet. The participant’s feet were placed inside two compartments within the wooden frame, and the 4 distracter lights were placed near the lower or the upper elevation (sole or the upper part of the feet) of both feet.27. A similar pattern of results emerges when the tactile and visual stimuli are on the same body side in comparison with stimuli that are on the opposite side of the body28. In fact, the two stimuli presented on the same side are processed as being inside the PPS around that body part, while when these are on the opposite side  the visual stimulus is processed as being outside the PPS of that body part30.The tactile and visual stimuli were presented almost simultaneously. The task involved judging the location of the tactile stimulus while ignoring the visual stimulus (i.e. the distractor lights). The stimuli were either congruent (i.e. with the tactile stimulus on the index finger and the distractor light on the upper part of the foot or incongruent (i.e. with the tactile stimulus on the thumb and the distractor light on the upper part of the foot). The difference in reaction times to the tactile stimuli in the congruent as compared to the incongruent condition is known as the Crossmodal Congruency Effect (CCE). The CCE is greater when the visual stimuli are within the participant’s PPS and smaller when they are outside the PPS22 was previously used by us in a study on SCI participants with complete paraplegia21.The CCE paradigm21 suggesting that the lack of sensory and motor functions affecting certain specific body parts impacts the representation of the space surrounding those parts. Significantly, the same study also showed that 15 minutes of passive leg mobilisation were enough to restore the PPS around the feet. However, to date, the mechanisms underlying the recovery of PPS remain unexplained.The results of that study indicated that there was a shrinkage in the representation of PPS around the feet of the participants affected by SCI and a normally preserved representation of hand-related PPS31), may be at play. For example, the recovery may be influenced by visual feedback relating to the body. Indeed, in the previously mentioned study concerning PPS in SCI21, passive mobilisation was administered while the participants were looking at their legs. This visual input might thus induce a sort of “illusion” of movement that in some way impacts on the individual’s PPS representation. Another hypothesis is that in the absence of below-lesion afference, somatic sensations coming from the body parts above the lesion level become particularly developed in cases of SCI. This may help individuals to rebuild their body and around-body representations during mobilisation.A variety of factors, ranging from feedback relating to the deafferented/deefferented body parts to interoception . This is based on sensory signals coming from body parts above the lesions . When associated with passive movement, the contribution of interoception to body awarenessThe aim of this study is to comprehend the mechanisms that underlie the recovery of PPS following passive mobilisation. All of the abovementioned components  are therefore considered. Our approach may contribute towards a better understanding of the role of the body in cognition and the interaction between bodily information and space representation.By means of Immersive Virtual Reality (IVR), we aimed to dissociate the impact of Motion and Visual information during passive mobilisation. In order to do this, four conditions were considered, recorded and analysed. These were related to whether the motor and visual information that the participants received was congruent or non-congruent: 1 - Vision: Mobilisation, Motion: Mobilisation; 2 - Vision: Mobilisation, Motion: No Mobilisation; 3 - Vision: No Mobilisation, Motion: Mobilisation and 4 – Vision: No Mobilisation, Motion: No Mobilisation. For each condition, data from the PPS assessment were compared to the PPS results in the baseline and follow-up assessments. Furthermore, the influence of autonomic, interoceptive signals and interoceptive awareness were recorded and considered in the main analyses. The experiment was divided into two sessions. In Session 1, we investigated the effects of showing a video in which the participant could see a pair of legs being passively mobilised (Vision: Mobilisation), either accompanied by actual passive movement (Motion: Mobilisation) of their legs or without any actual movement (Motion: No Mobilisation). In Session 2, we studied the effects of the participant being shown a pair of immobile legs (Vision: No Mobilisation) either with the passive mobilisation of the participant’s legs (Motion: Mobilisation) or without (Motion: No Mobilisation) Mobilisation. Behavioural and physiological measures were recorded in both sessions.The participants were divided into three groups: i) people who had suffered SCI with Complete lesion and Paraplegia (CP); ii) people who had suffered SCI but with Incomplete lesion and Paraplegia (IP) and iii) a group of gender and age matched healthy controls .There were 19 CP participants in total , with 9 of them taking part in both sessions. The IP group comprised a total of 23 participants , with 5 of them taking part in both sessions. The clinical and demographic data of the SCI participants are shown in Table A power analysis was executed with a moderate effect size (Cohen’s f = 0.25) to achieve a power of 80%. The suggested sample size was 13.64.21  was used to show the IVR videos. This was removed in the other phases. The videos were three-dimensional, filmed from an egocentric, first person perspective and lasted 2 minutes. One video  was administered after the task. The BPQ is a self-reporting measure of body awareness and autonomic reactivity. The items refer to the autonomic nervous system and investigates the individual’s awareness of information coming from their body concerning the status of their organs and tissues . This is designed to quantify a person’s individual interoceptive sensibility35. Items n. 12, 14 and 36 were removed since they investigate sensations which the CP participants were maybe unable to feel (i.e. the urge to urinate or defecate and the feeling of fullness in the bladder).The participants in the Complete SCI group were unable to feel either their legs or feet both in the passive motion condition and in the immobile condition. Nevertheless, their sensitivity to their internal state might have resulted in alternative sensations related to the perception of motion. In order to check for this variable and for the participants’ subjective reactions to the experimental manipulation, the Body Awareness subscale of the Body Perception Questionnaire . In this task, the participants held a tactile stimulator in each hand, with the index finger and thumb of each hand positioned on the top and the bottom of the device respectively. Various different combinations of tactile stimulation and visual distractor were administered in each trial. The tactile stimuli  were administered either to the upper or lower parts of the stimulation devices (i.e. index finger or thumb respectively) and on either the left or right stimulation device (i.e. the left or right hand). The visual distractor stimuli (3 × 50 ms separated by 50 ms gaps) were displayed close to one of the two foot compartments on the wooden frame, either on the upper or lower part of the compartment . The CCE indexes are increased when the visual distractors placed near the feet and the tactile stimulation to the hands are administered on the same body side.In Session 1 (Vision: Mobilisation), the participants sat in a comfortable position on a chair equipped with armrests and a reclining backrest see Fig. . The wooThe participants were requested to look at the fixation LEDs on the wooden frame on the floor and, as quickly as possible, verbally report where they felt the tactile stimulus on their hand (i.e. “top” or “bottom”), irrespective of which hand the stimulus was administered to. They were told to ignore the visual distractors. The verbal response options had been selected with a view to avoiding the presence of fricative consonants in order to facilitate the participants’ enunciation thereby avoiding prolonged response times which would render the RT computation difficult. There are fricative consonants in the Italian words for top (“sopra”) and bottom (“sotto”) and thus the response options chosen were the sounds: i) “TAH” when the tactile sensation was felt on the index finger (top position) and ii) “TOH” when it was felt on the thumb (bottom position), irrespective of whether the stimulus was on the left or the right hand.In order to ensure that the participants focused on the centrally positioned LEDs, they were instructed to say the word “luci”  whenever they saw the green LED flashing , and to guarantee that they paid attention to the tactile stimuli, false stimulations were administered. These false stimulations consisted of a distractor light on the wooden frame which flashed without tactile stimulation. In these cases, the response was expected to be “niente” . The PPS task consisted of 56 trials in the Baseline and at the Follow-Up blocks . No video was shown or any passive motion administered in these two conditions.In the other blocks (i.e. from the second to the seventh block), during Session1 (Vision: Mobilisation), a head mounted display device showed a 2 minute video in which the participants watched a pair of legs being passively moved by two people. While they observed the video, the participants’ own legs (which they could not see) could either be synchronously moved , with the difference that the video shown in the second to the seventh blocks depicted immobile legs . This was done to ensure that all of the participants had understood the task correctly. Moreover, all the reaction times of less than 100 ms were removed since it takes at least 100 ms for human beings to detect a visual target39, or to detect a tactile stimulus to the hand while focusing on a foot40. Inverse Efficacy (IE) indexes were then computed by dividing reaction times by the proportion of correct trials for each block and each participant . All the values outside the range [mean ± 2.5 SD] were considered outliers and were thus removed  indicates a range between ≥0 and <1). The outliers constituted 3.12% of the original data.Reaction times from the PPS task around the feet were extracted from audio files (WAV format at 8000 Hz) by means of a custom-made RThe IEs were then used to estimate the CCE. In order to take into account any variances in the entire data set, we used a specific Bayesian model .As the Movement Verbal Feedback was a binomial yes/no response, this was transformed to indicate accuracy .41.The physiological measurements were extracted from the signals recorded during the 2 minutes videos by means of the Acknowledge software ver. 4.2 . The Skin Conductance Signal  was low band filtered at 40 Hz, and the SCL values were obtained by subtracting the minimum value from the maximum value for each video. In order to have a physiological signal which was representative of the parasympathetic system, we computed the natural logarithm of the Respiratory Sinus Arrhythmia (RSA) from the pulse pressure wave and respiration signals. RSA gives a measure of the vagal tone as it reflects any variations in heart rate during inspiration and expiration45. Bayesian multilevel hierarchical models were used in all of the analyses47 (for an example in the frequentist context see48). This was done in order to take into account any within-subjects random variability that might have a confounding impact on the fixed, experimentally manipulated factors which address statistical population effects. In addition, since some of the participants took part in both experiments, data from both experiments were analysed in a single analysis, and the random intercepts and slopes were computed for each participant. As a result, individual deviations from the fixed factors for those participants who performed in both sessions were taken into account. For each analysis and for each factor, we computed the Bayes Factors (BF1049) via the Kuo and Mallick algorithm50. In order to interpret the results, we followed the seminal recommendations by Raftery51 who reports a BF10 within  and were thus not considered in the subsequent analyses.Strong effects for Laterality (BF10 = 7.85) and for the conditions where visual and motor stimulation were congruent [i.e. “Vision: Mobilisation – Motion: Mobilisation” (BF10 = 54.56) and the “Vision: No Mobilisation – Motion: No Mobilisation” (BF10 = 124)]. In contrast, in the “Vision: No Mobilisation - Motion: Mobilisation” condition only weak support for the alternative hypothesis was found (BF10 = 1.64) and in the “Vision: Mobilisation - Motion: No Mobilisation” condition there was positive evidence supporting the null hypothesis, namely the absence of a PPS representation (BF10 = 0.08).Post-hoc analyses of the Group:Condition:Laterality interaction confirmed the presence of a PPS representation around the feet in the C group in the “No Stimulation” condition (BF10 = 15.13) and “Vision: No Mobilisation – Motion: Mobilisation” (BF10 = 20.28) conditions, while in the “No Stimulation” (BF10 = 0.04), “Vision: Mobilisation - Motion: No Mobilisation” (BF10 = 0.15) and “Vision: No Mobilisation -Motion: No Mobilisation” (BF10 = 0.329) conditions there was positive or strong evidence supporting the null hypothesis, suggesting a lack of PPS representation.In the case of the CP group, there was only evidence of a PPS representation around the feet in the conditions where the passive movement was actually administered. In fact, PPS was present in the “Vision: Mobilisation - Motion: Mobilisation” (BF10 = 22.26) and “Vision: No Mobilisation – Motion: Mobilisation” (BF10 > 150) conditions. Nevertheless, in this group, PPS was also recorded in the “Vision: Mobilisation – Motion: No Mobilisation” condition (BF10 = 6.81), while in the “No Stimulation” condition there was strong evidence supporting the null hypothesis , and in the “Vision: No Mobilisation – Motion: No Mobilisation” condition there was only weak evidence supporting the null hypothesis (BF10 = 0.34). See Fig. The results for the IP group showed a similar pattern to the CP group, with strong evidence of the presence of a PPS representation around the feet in the “Vision: Mobilisation – Motion: Mobilisation” BF10 = 2.26 and In this analysis, the fixed effects were Group  and Condition , and the interaction between them. The within-subject Condition factor was also used as random slopes which was grouped among participants.10 > 150), showing that there were differences in the accuracy of Movement Verbal Feedbacks between groups and conditions, but only weak, positive evidence supporting the null hypothesis for the Group  and Condition  factors.This analysis showed very strong evidence supporting the alternative hypothesis for the Group:Condition  in the “Vision: No Mobilisation - Motion: No Mobilisation” condition.The CP group performed perfectly in the “Vision: No Mobilisation - Motion: Mobilisation” condition (42/42), but made mistakes in the “Vision: Mobilisation - Motion: Mobilisation” (34/42), “Vision: Mobilisation - Motion: No Mobilisation” (37/42), and “Vision: No Mobilisation - Motion: No Mobilisation” (38/42) conditions. In fact, while the comparison between the “Vision: No Mobilisation - Motion: No Mobilisation”, “Vision: Mobilisation - Motion: Mobilisation” and Vision: Mobilisation - Motion: No Mobilisation” conditions revealed strong, positive evidence supporting the null hypothesis (namely the absence of a difference between conditions), the comparisons with the “Vision: No Mobilisation - Motion: Mobilisation” condition all showed strong or positive evidence supporting the alternative hypothesis (presence of differences between conditions). Thus, in the condition where CP participants saw immobile legs but their own legs were actually being moved, they were better at detecting movement.Conversely, the IP group performed perfectly in the “Vision: Mobilisation - Motion: Mobilisation” condition (42/42), while they made mistakes in the “Vision: No Mobilisation - Motion: No Mobilisation” 36/42) “Vision: Mobilisation - Motion: No Mobilisation” (36/42) and “Vision: No Mobilisation - Motion: Mobilisation” (33/42) conditions. Comparisons between the “Vision: Mobilisation - Motion: Mobilisation” and “Vision: Mobilisation - Motion: No Mobilisation” conditions , and between the “Vision: Mobilisation - Motion: Mobilisation” and “Vision: No Mobilisation - Motion: Mobilisation” conditions revealed positive, strong evidence supporting the alternative hypothesis . The comparison between their performance in the “Vision: Mobilisation - Motion: No Mobilisation” and in “Vision: No Mobilisation - Motion: Mobilisation” conditions  showed weak evidence supporting the null hypothesis (therefore an inconclusive tendency towards a lack of differences between the conditions). See Fig. 6/42 “VisLastly, we checked in order to ensure that the performances were better than random guesses using the model in Table 10 = 4.07) and between Condition and RSA (BF10 = 8.13). The SCL covaried with the PPS representation as measured by CCEs in the “Vision: No Mobilisation - Motion: No Mobilisation” condition (BF10 = 75.92), and this covariation was positive, with greater CCEs linked to higher SCL responses . In the other conditions, the BF10 were within [0.70 ÷ 1.44], showing only weak, negligible evidence. The RSA positively covaried with CCEs in the Vision: No Mobilisation - Motion: Mobilisation condition , that is, greater CCEs were linked to higher RSA responses . In all of the other conditions, BF10 was within [0.75 ÷ 1.5], showing weak, non-conclusive evidence.The analysis of the relationship between the CCEs and the physiological data involved Group , Condition , the Skin Conductance Level (SCL) and the Respiratory Sinus Arrhythmia (RSA) covariates and their interactions. The random factors considered were the same as those used in the behavioural analysis, that is, Condition and the number of blocks in z-scores, grouped for each participant. There was positive evidence supporting the alternative hypothesis in the interactions between Condition and SCL (BF10 = 0.24), between Group and RSA (BF10 = 0.27) and in the main effect Group (BF10 = 0.19).Finally, positive evidence supporting the null hypothesis was found in the 3-way interaction between Group, Condition and SCL (BF10 = 305.12). In the case of Condition (BF10 = 7.35) and BPQ, there was positive evidence supporting the alternative hypothesis (BF10 = 3.06). In particular in the IP group, BPQ positively covaried with PPS representation (CCEs) in the “Vision: No Mobilisation - Motion: Mobilisation”  and “Vision: Mobilisation - Motion: No Mobilisation” conditions . This suggests that, in cases of incomplete lesions, greater interoceptive sensibility is associated with a better representation of PPS in conditions where motor and visual stimuli are incongruent.An evaluation of interoceptive awareness was made by analysing the Body Perception Questionnaire scores in relation to the representation of PPS around the feet following the same structure as that used in the previous analysis. There was very strong evidence supporting the alternative hypothesis in the Group:Condition:BPQ interaction (BF10 = 4.88) where cases of a lower degree of lesion – less deficit were associated with a better PPS representation around the feet . The other factors were not correlated with PPS representation, (BF10 were within [0.64 ÷ 2.03]).The clinical scores for AIS and NLI were covaried with the PPS representation around the feet by taking into consideration the same fixed and random effects as those used in the previous covariation analyses. Positive evidence supporting the alternative hypothesis was found only in the NLI covariate . Conversely, in the group of controls, the mismatch between visual and motor feedback disturbed the representation of PPS.When explicitly asked, the CP participants were able to accurately identify  the presence of movement (even though it was not visible), suggesting that some residual perception of movement may persist even after complete SCI. Nevertheless, our results indicate that the processes underlying the recovery of PPS are, at least in part, implicit and probably linked to interoceptive functions that may be less affected by the lesion.55. Although we did not assess this aspect directly, the attentional component was kept constant in our experiments, as also confirmed by the absence of any differences in performance between baseline and follow-up. We can thus consider our measure of CCE as a specific index of PPS representation. We found a reduction in PPS associated with SCI-related deafferentation/deefferentation in both CP and IP participants. In fact, in the baseline and follow-up conditions , only the C group showed a CCE (as an index of PPS representation). These data are in line with the observation made by Bassolino and colleagues56 that PPS is reduced as a result of a lack of ability to move. They also reinforce the data from our earlier study which provided evidence that paraplegic participants with complete lesions showed a reduction in PPS around their feet but not around their hands21. Interestingly, it seems that this is not true for upper-limb affordances. Indeed, in a study on 10 patients with heterogeneous SCIs, Sedda and colleagues57 did not find faster affordance judgements on the reachability of objects in peripersonal space, while in the control group they found the effect of PPS judgement.It has been suggested that CCE is modulated by attentional processes relating to response mapping competition58 and with the actual possibility of moving. Recently, a broader definition proposes that “PPS is a network of body-part-centered representations responsible for the actions toward, and in avoidance of, objects and other living entities, including people.”59. In fact, PPS representations can be modulated by several top-down variables, such as social contexts60, defensive components62, anxiety levels64, body representation66 and non-defensive action capabilities68. This complexity suggests a multilevel network organisation69.PPS is defined as the “space in which actions are performed”, and PPS representation is considered to be associated with action representation22. Most important for the aim of this study is the fact that separate PPS representations anchored to the hand, face, or trunk do in fact exist, but they are not independent of each other70.PPS is usually assessed by applying visual and tactile stimuli to the same body part. However, there is some evidence which indicates that the visual component of PPS is plays a stronger role in the detection of whether a stimulus is within or outside the PPSThe present study focuses on the relations between PPS and body representations.The data collected confirm that lower lesions (i.e. fewer deficits) are associated with a greater representation of PPS around the feet in both the CP and IP participants, indicating that changes in the representation of space are directly related to sensory-motor deficits.28) and passive movements can improve PPS representation. This is in line with an observation that, in upper limb amputees, the representation of PPS around the hands shrinks to the stump but can be enlarged by a wearable prosthesis67. However, until now it has not been totally clear whether this effect is due to (i) visual information about body shape, (ii) the possibility of actually performing movements using a prosthesis or (iii) an interaction between these two mechanisms. It was observed for the first time in an earlier study involving complete paraplegics21 that the plasticity of the PPS around the feet is mainly linked to the movement of the lower limbs (even passively) rather than to body shape. It was found that (probably latent) bodily information plays a role in modulating PPS representation. In the present study, the performance of the healthy controls suggests that consistency and congruency in the stimuli coming from different sensory modalities are needed in order for PPS representation to be preserved. In effect, our controls show an intact PPS representation in all of the conditions in which visual and motor information are congruent. However, visuo-motor incongruency induced a subjective mismatch between what was seen and felt that ultimately lead to an impairment of the lower-limb related PPS representation. In situations involving incongruent cross-modal stimulation, our cognitive system tends to integrate the information in a single percept72 that may lead to the subject in question interpreting the experience as “my legs are not really moving”. Another possibility is that the sight of one’s own legs being moved without concomitant proprioceptive feedback (and vice-versa) results in the perception of an impossible action. This might inhibit the representation of PPS as a consequence of the lack of activation of the networks which are usually involved in the internal representation of actions75. Since the same cortical networks are also activated in PPS representation , it is possible that observing an impossible action interacts with and inhibits PPS representation.The results of this study also show that both active  led to the recovery of PPS. In fact, the CP participants recovered PPS representation in both the “Vision: Mobilisation - Motion: Mobilisation” and “Vision: No Mobilisation - Motion: Mobilisation” conditions. This represents an unexpected result as in cases of complete lesion, not only motor but also somatosensory pathways are completely interrupted, and no sensory information should reach the somatosensory cortex. However, the CP participants correctly reported the presence of movement in their subjective verbal responses, in particular in the “Vision: No Mobilisation - Motion: Mobilisation” condition. Moreover, during the informal interview after the experiment, these participants often said that after the injury, they had progressively learned to pay attention to new sensations (e.g. similar to chills) from the above-lesion body parts and from inside their body (interoception) that signalled their lower limbs were being moved. Thus, the improvement in PPS representation in this group in the conditions where movement was induced may be due to these above-lesion signals and their integration with higher level cognitive functions .77, namely, lesions which are complete from a functional and clinical point of view, but that maintain some distal forms of brain-body connections which the patient is unaware of.Another possibility is that this recovery is an effect of so-called “discomplete lesions”In the present study, the paraplegics with incomplete lesions showed a recovery of the representation of PPS around their feet in all of the conditions where there was manual Mobilisation and/or a visual component involving movement . This is not surprising with regard to the Vision: No Mobilisation - Motion: Mobilisation and Vision: Mobilisation - Motion: Mobilisation blocks as these patients have residual somato-sensory functions.The recovery of PPS in the Vision: Mobilisation - Motion: No Mobilisation condition is however more difficult to explain. Indeed, the IP group seemed to be sensitive to the incongruence between visual and motor signals, but in an opposite direction with respect to the group of controls, that is, PPS representation in incongruent blocks was reduced in Controls and improved in IP. We suggest that incongruent conditions had the effect of making the IP group shift their attention from the movement (which was absent) to bodily information. This may have led to an increased representation of their body in the space around them with indirect effects on PPS representation. In effect, only in these blocks was PPS representation recovery correlated positively with the degree of interoceptive awareness (as measured by means of the Body Perception Questionnaire).79, while the orthosympathetic route is influenced by lesions higher than T578. In contrast, disorders in autonomic responses have been reported after cervical lesions81. Our expectation was that, as a consequence of the lack of somatosensory input, there would be an increment in the sensitivity of the paraplegic participants to signals from inside their body which would lead to an increase in interoceptive responses. However, we did not find any difference in autonomic responses between the three groups. Furthermore, as part of a series of additional analyses, the SCI participants were divided by lesion level (<T5 or ≥T5) and completeness, but there were no differences in either RSA or SCL.Data regarding the changes in autonomic responses after SCI have been to date meagre. It has been found that autonomic parasympathetic control functions are not impaired by paraplegia82. Furthermore, our results may be in keeping with reports of the activation of the parasympathetic system during passive movement (Vision: No Mobilisation - Motion: Mobilisation)83.It is worth noting that the experiment was not designed to modulate SCL or RSA responses. Indeed, the stimuli depicted either a slow, smooth passive motion in the lower limbs, or no motion at all, accompanied by constant contact with the two experimenters’ hands on the participant’s thighs even though the faces of the experimenters were always out of sight  which positively covaried with PPS representation in the IP participants in the incongruent conditions. This suggests that SCI patients with a higher interoceptive awareness might also refer to this typology of information when faced with conflicting, discrepant visuo-motor stimuli.84 and Noel et al.85. After 15 minutes of audio-visual deprivation, PPS representation becomes ill-defined. Although the participants’ responses in interoceptive accuracy were different (i.e. some indicating improvements and some worsening), the study found a correlation between PPS representation and the modulation of interoceptive accuracy85. Our results are consistent with these data indicating that individuals with better interoception show greater modulations of PPS representations.The connection between PPS representation and interoception was previously observed in healthy participants by Ferri, Chiarelli, Merla, Gallese, & Costantini86, there is evidence indicating that paraplegic patients may suffer from a dramatic reduction in motor imagery capacities91 and in their ability to discriminate biological motion . This is true even when they are aware of their motor impairment92. These deficits have been found to be topographically specific as they involve actions that regard the paralyzed body parts but not those which are habitually executed by the upper body parts21. It is also worth noting that while a recent study reported a deficit in learning new motor sequences7, other studies have shown that SCI patients acquire new post-lesional motor abilities and that this also leads to specific expertise in visual action discrimination8.Cognitive studies on SCI participants have demonstrated that body-brain disconnection (without brain lesions) leads to a somato-topographically specific reorganisation of several cognitive functions. Action representation alters after SCI. Although people with SCI report that they walk in their dreams21. All in all, our study supports the notion that cognitive functions such as the representation of space are grounded on sensory-motor functions and bodily signals. In fact, if cognitive functions such as space representation was only based on high order, symbolic systems, no changes would be recorded after a lower level, spinal body-brain disconnection.An interesting result from the present study is that not only action and body perception but also the representation of PPS were modified in the SCI participants according to topographically specific rules94, with a probable role of deficit compensation and re-adaptation. Importantly, however, maladaptive plasticity leading to neuropathic pain and corporeal illusions96 has also been reported. These negative effects may impact on any approach to treatment based on EEG rhythms99. Thus, it is extremely important to ascertain whether there are any changes in cognitive functions after a sensory-motor deficit and to utilise tasks that can help people with SCI to maintain cognitive functions and contrast maladaptive plasticity101. Our results show that simple passive motion for only 2 minutes may lead to the functional recovery of space representation. Although this recovery did not last until the Follow-Up assessment, it may be the case that more focused, sustained protocols would result in more lasting outcomes. This would support the results of a number of previous studies which have indicated that motor rehabilitation has positive effects on cognition103.Widespread reorganisation of the brain has been documented after SCI106. These modifications probably originate from the widespread neuroplastic modifications that occur after spinal cord lesions111 in neural regions underpinning both bodily representation and the mapping of peripersonal space112. These neuroplastic changes are mediated by multiple sources of information coming from the body. A perfect balance between these appears to be necessary to guarantee thecorrect representation of PPS.Overall, our results support the hypothesis that changes in cognitive functions follow conditions of deafferentation and deefferentationNevertheless, residual bodily information seems to assist the recovery of PPS in SCI. This supports the notion that bodily information is fundamentally important for the representation of PPS thus providing evidence supporting the notion of embodied PPS.Supplementary Materials.


Globally, malnutrition underlies 45% of under-5 s mortality, mainly from potentiating common infections such as diarrhoea and pneumonia. Malnutrition as a public health problem is not evenly disbursed because of disparities in food insecurity and health, and children commonly suffer recurrent episodes of opportunistic infections. We aimed to understand better the spatial and temporal structure of multiple paediatric hospital admissions associated with malnutrition-related illnesses. This paper aimed to investigate the spatial-temporal variations in malnutrition-related recurrent morbidity of children under-5 years from the Kilifi County in Kenya between 2002 and 2015.The study included data from children under-5 years old who had more than one admission to a rural district hospital in Kenya within the Kilifi Health and Demographic Surveillance System (KHDSS). The primary outcome was a malnutrition-related admission, based on wasting (WHZ < -2) or nutritional oedema. Individual, household and environmental level covariates were examined as exposures. We first fitted a SARIMA model for the temporality, and the Moran’s Index affirmed spatial clustering in malnutrition admissions. Kulldorf Statistics using SaTScan were applied to detect hotspots. Then, bivariate analysis was done using repeated values tabulation and analysis of covariance (ANCOVA). Inferential analysis was done using a mixed effect multivariable negative-binomial regression model, adjusting for spatiotemporal random effects.A total of 2821 children were admitted more than once, giving a total of 6375 admissions. Of these 6375 admissions, 1866 were malnutrition-related, and 3.9% (109/2821) of the children with repeat admissions died. There was a seasonal pattern of re-admissions, peaking from May to July over the years. Hotspots were found in both the Northern and Southern areas of the KHDSS, while the areas near Kilifi Town were least affected. We found that disease severity was most likely associated with a malnutrition re-admission to the hospital.Disease severity was strongly associated with admission with malnutrition but its effect reduced after adjusting for the spatial and temporal random effects. Adjusting for clustering in space and in time  in models helps to improve the understanding of recurrent hospitalisations involving malnutrition.The online version of this article (10.1186/s40795-019-0296-5) contains supplementary material, which is available to authorized users. Globally, malnutrition is a cause of 45% of childhood mortality, predominantly by potentiating morbidity due to common infections such as diarrhoea and pneumonia , 2. Sub-Undernutrition is a result of many determinants and not solely related to food insecurity. In Kenya, fundamental immediate and underlying causes of undernutrition have been identified to vary across the different age groups of 0–24 months and 25–59 months. Child’s birth size, breastfeeding patterns and acute morbidity experience have more impact on the younger age group compared to the underlying factors affecting the older age group , 9, 10. Because malnutrition is due to a complex interaction between different components, it is difficult to understand in a way that helps extend preventive and treatment strategies from current vertical interventions , 11. MalSecondary data analysis was conducted retrospectively on observational data from 2002 to 2015 of recurrent pediatric ward admissions to Kilifi County Hospital (KCH) in Kenya, including household data from a demographic surveillance. Kilifi County is located in the Coastal region of Kenya  < − 2 or the presence of oedema at admission. Oedema is a clinical sign of undernutrition which may include swelling of the feet and skin . The WHZThe explanatory variables were selected with the guidance of the UNICEF malnutrition conceptual framework . Child lA team of field workers were trained on using the KHDSS surveillance system to do the matching of the data on real-time admissions. Each field worker has access to the web-based system following the matching procedures of the patients explained elsewhere . QualifiTo explore the temporal patterns, the data were defined as a regular time series data using monthly time points of counts of malnutrition-related admissions. Augmented Dickey-Fuller unit-root was used to test for stationarity; the null hypothesis is there is no seasonality trend in the model. The stationary series was then examined on the Auto-Correlation Function (ACF) for the MA lags, and Partial Auto-Correlation Function (PACF) for the AR lags. This was followed by fitting combined Autoregressive integrated moving average (ARIMA) and the seasonal component to form the SARIMA models detailed in Additional file Seasonality was determined using the ACF and PCF with pointwise confidence intervals based on Bartlett’s formula. The Portmanteau test was used to confirm the significance of the seasonality. The Portmanteau test null hypothesis is that there is no serial correlation using the white noise under Pearson’s Chi-Square statistic . This waI > E(I) shows a positive autocorrelation implying clustering and I ≈ 0 shows no spatial autocorrelation and I < E(I) shows a negative autocorrelation implying a dispersed neighbouring value, where E(I) is the normally distributed population mean.The Global Moran Index was used to estimate the measure of sub-location spatial autocorrelation or randomness based on the observed cases of malnutrition admission. A distance matrix of the sub-locations centroids was used to calculate the Moran’s index , 26. TheA window scanning over space was done comparing the observed versus the expected over the country using the SaTScan software. SaTScan applies the Kulldorff spatial scan statistic which imposes a circular window and calculates the likelihood of observing the events inside and outside the study area. The circle with the maximum likelihood is defined as the most likely cluster , 28. ThiOur outcome was defined as the counts of malnutrition-related admissions, reflecting the burden and distribution of malnutrition and pattern of hospital utilisation. The data had a repeated event structure for the individuals over time. For the bivariate analysis, panel data tabulations were applied to investigate the between-individual variation and within-individual variation. This helps to understand the individual systematic difference and differences between individuals over time.Repeated measures analysis of covariance (ANCOVA) was applied to compare the differences between the continuous predictors and the outcome.The malnutrition related morbidity was observed to follow a negative binomial distribution. Negative binomial is a generalisation of the Poisson distribution used to provide better epidemiological estimates of factors associated with malnutrition with repeated data , 30. SecMultivariable analysis was done using non-spatial, spatial and spatial-temporal models for admission specific and environmental variables with sub-location information.We used the adapted Stochastic Partial Differential Equations (SPDE) in Integrated Nested Laplace approximation (INLA) for the spatial-temporal model, and the Markov chain Monte Carlo (MCMC) with Metropolis-Hastings algorithms approaches in OpenBUGS software from Medical Research Council (MRC) Biostatistics Unit  for spatOver the 14 years, 2821 children under-5 years had at least two admissions in KCH, totalling 6375 admission events including the initial admission. Malnutrition-related admissions were identified in 1054/2821 (37% of children ever admitted) of the children. Of the 1054 children ever admitted with malnutrition, 76% (percentage within) of their re-admissions were malnutrition-related .The AutoCorrelation Function (ACF) and Partial autocorrelation Function (PACF) of the transformed series using data from 2002 to 2015 showed peaks at different lag periods as shown in “Additional file p < 0.001) Portmanteau test for white noise. Different peaks are observed, but in general, the malnutrition-related admissions decline over the period. The tables in “Additional file Significant serial correlation  of months between May and July were observed for children admitted with malnutrition as shown in Fig. p-value< 0.001, sd = 0.029) was observed showing a positive autocorrelation implying clustering. Based on the calculated Global Morans Index, we rejected the zero spatial autocorrelation hypothesis. The Kulldorff spatial scan statistic using SaTScan showed hotspots in the Northern and Southern areas of Kilifi urban area where the main hospital is based. Areas closer to the main hospital and Kilifi town were observed as cold spots of malnutrition-related admissions as shown in Fig. A Global Moran’s I of 0.1  approach and Multilevel modelling are in “Additional file As shown on Table In general, the spatial-temporal models had a lower DIC compared to non-spatial models but with extratemporal parameters. Bayesian spatial-temporal model using R-INLA was the model with the best fit with a DIC = 10,656.28 (pD = 28.41) which was the lowest compared to the spatial model which had a DIC =10,982.20 (pD = 21.46).The spatial-temporal model showed consistent results with the non-spatial model except for the severe disease which changed. The final spatial-temporal negative binomial model was considered the “model of best fit” which caters well for over-dispersion and spatial-temporal confounding.Adjusting for the spatial-temporal effects and other factors, males were less likely to have malnutrition re-admission (RR = 0.88 (Cr.I = 0.80–0.96) compared to the females. In the non-spatial model, the increase in the number of severe diseases increased the risk of malnutrition readmission. On the same note, the spatial-temporal effects, the levels 2 and 3 of severe diseases were insignificant. The environmental variables were not significantly associated with malnutrition re-admission in both the multilevel and spatial-temporal models.Posterior temporal estimates of readmission were displayed on structured maps shown in Fig. This study describes the spatial-temporal admission of a group of children with repeated malnutrition-related re-admissions to Kilifi County Hospital admissions between 2002 and 2015. A seasonality of malnutrition-related re-admissions was observed, with peaks occurring in July when the rainy season was mainly observed. Emelda et al. reported a significant decline in malaria admissions at many hospitals in Kilifi. They reported this was observed against a background of elevating or regular non-malaria admissions and impervious to long-term rainfall throughout the surveillance period . In SomaSevere diseases, days of admission and rainfall were fit as covariates with malnutrition readmission as the outcome on our model. Severe disease was defined as the presence of either gastroenteritis, Lower respiratory tract infection (LRTI), blood and Cerebrospinal fluid (CSF) culture positive, positive malaria test examined by microscopy and fever or meningitis. In Somalia and Malawi, infections and geographical factors, including the Enhanced Vegetation Index were observed as critical drivers of malnutrition , 38. SimIn contrast to EVI, rainfall seasonality was associated with malnutrition admission which is similar to the time series analysis done by Karuri et al. that reported a seasonality during rainfall peaks for malaria admissions in the Kenya Coast . This waSabrina et al. in 2014 review recommended the importance of combining spatial and temporal components in understanding the compounded phenomenon of malnutrition . Though,This study demonstrates that it is feasible to map episodes of repeated admission to hospital with malnutrition related morbidity at high spatial resolutions. The work has also utilised recently developed statistical tools together to develop spatial-temporal models that converged rapidly without loss of predictive accuracy. The contribution of infections provides a better understanding of the drivers of repeated admissions with malnutrition as a co-morbidity.Additional file 1:Appendix 1. Flow diagram of data inclusion and exclusion. (DOCX 22 kb)Additional file 2:Appendix 2. Time series and Bayesian model derivation. (PDF 144 kb)Additional file 3:Appendix 3. Monthly malnutrition related admissions in Kilifi County Hospital between 2002 and 2015. Appendix 4. Autocorrelation Function (ACF) and Partial autocorrelation function (PACF) of the SARIMA model. Appendix 5. Shifting hotspots and colsdpots from 2002-2015 using Kulldorff statistics in SaTScan. Appendix 7. Negative Binomial Bayesian fit Diagnostic Plots. (DOCX 4710 kb)Additional file 4:Appendix 6. R code for environmental data extraction and fitting the spatial temporal model. (DOCX 28 kb)


The linear RF has a number of inherent problems: it changes with the input, it presupposes a set of basis functions for the visual system, and it conflicts with recent studies on dendritic computations. Here we propose to model the RF in a nonlinear manner, introducing the intrinsically nonlinear receptive field (INRF). Apart from being more physiologically plausible and embodying the efficient representation principle, the INRF has a key property of wide-ranging implications: for several vision science phenomena where a linear RF must vary with the input in order to predict responses, the INRF can remain constant under different stimuli. We also prove that Artificial Neural Networks with INRF modules instead of linear filters have a remarkably improved performance and better emulate basic human perception. Our results suggest a change of paradigm for vision science as well as for artificial intelligence.The responses of visual neurons, as well as visual perception phenomena in general, are highly  Many scientists have come to accept this linear plus nonlinear (L+NL) formulation as a working model of the visual system2, and while there have been considerable improvements on, and extensions to, the standard model, the linear RF remains as the foundation of most vision models. The inspiration for Artificial Neural Networks (ANNs) came from early, classical models of biological neural networks, and for this reason they are also based on the linear RF, which is their cornerstone4.In vision science, the receptive field (RF) of a neuron is the extent of the visual field where light influences the neuron’s response. In the “standard model” of vision the first stage is a filtering operation consisting of multiplying the intensities at each local region of an image stimulus by the values of a filter (the weights of the RF), and summing the resulting intensitiesBut there are a number of problems that are inherent to considering the RF as having a linear form, of which we will highlight three.5. Adaptation constantly adjusts the sensitivity of the visual system to the properties of the stimulus, bringing the survival advantage of making perception approximately independent from lighting conditions while quite sensitive to small differences among neighboring regions7; this happens at very different timescales, from days and hours down to the 100ms interval between rapid eye movements, when retinal neurons adapt to the local mean and variance of the signal, approximating histogram equalization8. In this way, adaptation allows to encode neural signals with less redundancy, and is therefore an embodiment of the efficient representation principle10, an ecological approach for vision science that has proven to be extremely successful across mammalian, amphibian and insect species14 and that states that the organization of the visual system in general and neural responses in particular are tailored to the statistics of the images that the individual typically encounters, so that visual information can be encoded in the most efficient way, optimizing the limited biological resources.For all species, adaptation is a key property that any neural system must have; in particular in the human visual system it is present in all stages, from the photoreceptors in the retina all the way to the cortex16. It is visual adaptation which modifies the spatial receptive field and temporal integration properties of neurons depending on the input; in fact, under a basic L+NL formulation, adaptation simply means “a change in the parameters of the model”17.Due to the fact that the visual system is nonlinear, it can be shown that the linear RF can’t be a fixed, constant property of a neuron20.The conclusion is that the RF of a neuron is not determined by its biological wiring but has different sizes, preferred orientations or even polarity (ON/OFF) for different stimuli15; the fact is that vision models and ANNs use RFs that are constant, or they don’t have general rules, valid for all inputs, as to how the RFs or the nonlinearities should be modified depending on the stimulus.In vision models the RF is characterized by finding the linear filter that provides the best correlation between the visual input and some specific response data, be it psychophysical like brightness perception magnitude or neurophysiological like spike train timings, and for that data the model can normally achieve an excellent fit. But there is much evidence showing that model performance degrades quickly if any aspect of the stimulus, like the spatial frequency or the contrast, is changed2; in fact, the estimated linear RF changes with the basis set used21. Models based on the linear RF are not tests of how well the RF describes the neuron’s behavior, they have been obtained simply by assuming that the neuron performs a linear summation followed by a nonlinearity and then searching for the best-fitting L+NL model.The visual system is nonlinear, and therefore it has no basis functions. Vision scientists, although aware that neurons are nonlinear, model their responses by probing the visual system with a variety of mathematically elegant basis functions like spots, edges or gratings, which are ideal for analyzing linear systems but that for nonlinear systems have no particular meaning2. As experimental constraints impose to use just a tiny fraction of tests, the nonlinearities that are mapped are forced to be quite smooth. Therefore, nothing guarantees that the behavior of the visual system for all inputs, and in particular for complex stimuli as those given by natural scenes, can be characterized from studying a reduced set of stimuli.Studying neurons using the complete set of possible stimuli, which according to the theory would be the only way to characterize the system with total confidence, is a task of impossibly high complexity, requiring in the order of 22, showing instead that each thin branch of the elaborate dendritic tree of a neuron could be computing many different nonlinear combinations of its inputs24.Cortical neurons produce all-or-nothing action potentials instead of analog values, i.e. neurons are highly nonlinear, and more recent studies have rejected the hypothesis that individual neurons can be modeled as a linear RF followed by an output nonlinearity25, divisive normalization26, feedback connections27, neural-field equations28, nonlinear geometry21, nonlinear centre-surround interactions29, hierarchical models30, fitting ANNs to visual data31, or training ANNs to perform a high-level visual task32, to name some of the most relevant lines of research.The linear RF limitations in predicting neuron responses to complex stimuli have been known for many years, and a wide variety of approaches have been introduced to model the nonlinear nature of visual phenomena, e.g. nonlinear subunitsHowever, we must remark that all these approaches are still grounded in the notion of a linear RF. And what we currently have is that state-of-the-art vision models and ANNs, with their linear RFs, have very important weaknesses in their predictive powers.open: for natural images under given viewing conditions, there are neither fully effective automatic solutions nor accurate vision models to predict image appearance, not even in controlled scenarios like cinema theaters8. This is a very important topic for imaging technologies, which require good perception models in order to encode image information efficiently and without introducing visible artifacts, for proper color representation, processing and display.In visual perception and color imaging, the general case of the image appearance problem is very much 33; this is a key limitation of ANNs, with an enormous potential for causing havoc. Another example is that the classification performance of ANNs falls rapidly when noise or texture changes are introduced on the test images, while human performance remains fairly stable under these modifications34. The difficulty of modeling vision with ANNs is a topic that is garnering increasing attention36.In computer vision, some of the well-known and most relevant problems of ANNs can be described as a failure to emulate basic human perception abilities. For instance ANNs are prone to adversarial attacks, where a very small change in pixel values in an image of some object A can lead the neural network to misclassify it as being a picture of object B, while for a human observer both the original and the modified images are perceived as being identical2. That same year, Carandini et al.1 commented that the fact that the linear RF component per se, which they say is “the bread and butter of the standard model”, can explain just a small fraction of the response variance of cortical neurons, is a sobering realization that leaves little room for the optimistic hope that matters could be fixed adding more modulatory terms or nonlinearities. Confirming their assessment, fifteen years later the standard model  is just able to explain around 37.In neuroscience, back in 2005 the standard model was able to explain at the most a 38 states that the problem is not in the lack of data but in the lack of a proper conceptual framework with which to analyze the data: the main obstacle is the standard model itself, which rather than be revised needs to be discarded altogether as it is the wrong starting point.Olshausenx is expressed as a weighted sum of the input I at neighboring locations 39, yielding a nonlinear summation model:39, and suggest as a possible reason that a single nonlinearity Poirazi et al. mention that with the model of Eq.  predicti24.There are mechanisms that enable individual dendritic branches to act as nonlinear units23.There is feedback from the neuron soma to the dendritesIn this paper we introduce the concept of an intrinsically nonlinear receptive field (INRF), a single-neuron summation model that we define in this way:In the classical model, the linear RF response at location x, given by the convolution with the kernel g, i.e. In the INRF, as schematized in Fig. mentclass2pt{minimsummation alternative to the linear RF. The final neural output would be INRF followed by a nonlinearity such as rectification, divisive normalization26, etc.Our INRF model is a model of x, because it depends on the local average not be expressed as a weighted average of the input passed through some nonlinearity: the same point not shift invariant, and therefore it can not be expressed as a convolution. In other words, the INRF can’t be expressed as a usual two-layer neural network, and for this reason we say that the INRF is intrinsically nonlinear and fundamentally different from the traditional nonlinear subunit summation models pioneered by Shapley and Victor40, as those models do follow the regular L+NL paradigm.A key point about the INRF model derives from the fact that the nonlinearity R(x), and assume for simplicity that the kernel g is very narrow and can be considered like a delta function:x, R(x) adds up the contributions from all neighbors R(x) computes the difference between I(x), passes this difference through the nonlinearity x and R(x). If we wanted to express R(x) using filters, we would need to have a different filter for each neighbor x:J where the value at each location w, producing the response at each location x:for a single pointx in an image with N discrete locations (or pixels) would require w for R(x), plus another convolution with m), when the classical linear RF needs just one convolution for the whole image, of course. For example, a INRF in a one-megapixel image would require over a million linear filters to be expressed in L+NL form.In fact, it would be extremely inefficient to express the INRF in the conventional L+NL framework. To understand the problem, let us concentrate on the term of INRF with the nonlinearities, call it m, g are Gaussians, and kernel w is a sum of a wide and a narrow Gaussian as argued in41 for lateral inhibition in the retina, then we see that the resulting kernel k can have the form of a Difference of Gaussians (DoG) linear RF, a stalwart in vision science42.In common scenarios the INRF simplifies to a linear RF. For instance, if mentclass2pt{minim43 showed that the following partial differential equation (PDE) performs histogram equalization:x denotes the spatial location, the input signal I is in the range  and I. The steady state of Eq. (g (akin to a delta function), then Eq.  is better at reducing redundancy than normalization, and the INRF is biologically plausible unlike other mechanisms for redundancy reduction that are also derived from Barlow’s principle such as radial factorization45. In image processing, histogram equalization is a classical technique to improve image appearance and it has been used in a variational model for perceptual color correction46, whose gradient descent equation has a term that is a particular instance of the INRF model; this links as well our INRF formulation with the very effective color enhancement model of Rizzi et al.47.Thus, we can argue that the INRF approximates local histogram equalization and for this reason it’s an embodiment of the efficient representation principle. Therefore, modeling the RF of retinal ganglion cells with a INRF could explain how these neurons are able to perform a constrained form of histogram equalization in the very short time interval between rapid eye movements48, which have a long and thriving history of modelling cortical low-level dynamics28. It has recently been shown49 that the Wilson–Cowan equations are not variational, in the sense that they can’t be minimizing an energy functional. The simplest modification that makes them variational yields a model whose main term has the form of an INRF. This modified model accomplishes local histogram equalization, and predicts brightness perception phenomena better than the original Wilson–Cowan model.Furthermore, we can relate the INRF to another successful vision science theory, that of the Wilson–Cowan equationsThe INRF has a key property of wide-ranging implications, our results show: for several vision science phenomena where a linear RF must vary with the input in order to predict responses, the INRF can remain constant under different stimuli.20.A constant INRF can qualitatively explain the very puzzling neurophysiological data by which OFF cells in V1 turn into ON cells when the spatial frequency of the stimulus increases20 is quite striking because it suggests that the neuron changes the pathway it belongs to (ON or OFF) depending on the input, and modeling this phenomenon with a linear RF requires not just a minor modification of the RF with the input but a complete inversion of polarity of the RF as the spatial frequency changes.We recall that neurons in the ON pathway respond to stimuli above the average level, while neurons in the OFF pathway respond to stimuli below the average level. Under the standard model, the experimental data inthe same INRF responds to stimuli above the average; see Fig. 50.With the INRF formulation, on the other hand, we can use a fixed INRF and predict the data: when the stimulus spatial frequency is low the INRF responds to inputs below the average, and when the spatial frequency is high oid see “”, hintin51 as it’s very much dependent on the input. For instance, if in the experiment above the uniform surround is replaced by salt and pepper noise of the same average (example images are shown in the bottom row of the figure), the crispening virtually disappears, see Fig. Figure 42, and the second step is a INRF that models the responses of retinal ganglion cells, where kernels m, w are simple Gaussians, g is a Dirac delta and the nonlinearity We have found that the same INRF, i.e. using a fixed set of parameters for our model, can adequately predict how crispening happens with uniform backgrounds, see Fig. ons see “”.If, after the Naka–Rushton stage, one were to use the classical L+NL formulation with a Difference of Gaussians (DoG) kernel and a pointwise nonlinearity instead of the INRF, it would be possible to optimize its parameters so that the L+NL model fits the psychophysical data and predicts crispening for the uniform background condition, as shown by the red curve in Fig. I and a distorted version of it I),INRF-B.Another very interesting example is that of White’s illusion, a brightness illusion in which two identical gray areas are perceived as being different due to their disparate surrounds. Vision models based on a linear RF can reproduce the illusion, but they fail to match the psychophysical data when bandpass noise is added to the image“irradiation illusion”, a white square over black background is perceived as being larger than a black square over white background. This phenomenon can be reproduced with a L+NL model that changes with the stimulus57, but we are able to model this illusion simply with a fixed INRF, see Fig. We have been able to qualitatively reproduce as well the perceived spatial asymmetries for lights and darks using a constant INRF. In the m, a standard deviation of w, a value of g being a Dirac delta and the nonlinearity Remarkably, the results shown in Table All the visual perception experiments considered are instances of brightness perception, and the INRF units used could be thought as modeling retinal ganglion cells. Hence they are preceded by some sort of photoreceptor nonlinearity, but that is not a central part of the model, it is only required in this brightness context. An indication that this is not a crucial part of our model lies in the fact that different photoreceptor nonlinearities have been used for the four visual perception experiments see “”, yieldiFinally, we have tested the idea of modifying a convolutional neural network (CNN), replacing each of its linear filters by a INRF while keeping all other elements of the architecture the same and maintaining the number of free parameters, then training this new network and comparing the performance with the original CNN. We’ve done this experiment for an image classification task, using two architectures and the four benchmark databases that are regularly used in the literature. As shown in Table We have presented the INRF and demonstrated that it has a number of remarkable advantages with respect to the classical linear RF. The INRF, introduced as a physiologically-plausible single-neuron summation model that is consistent with more recent studies on dendritic computations, embodies the efficient representation principle and provides an explanation for the ability of retinal cells to perform very rapid histogram equalization. For several phenomena in visual neuroscience and visual perception, the INRF is shown to have the capability of remaining constant for different inputs while the linear RF must change with the stimulus, indicating that the INRF might actually be determined by the cell’s wiring. Furthermore, we have been able to find a set of parameter values with which a single INRF element  is able to qualitatively predict perceptual responses in situations as diverse as the “crispening” effect, appearance of visual distortions in an image quality database, White’s illusion and the irradiation illusion. This supports the argument that the INRF is an adequate form of RF to model visual phenomena, and suggests that, as with the linear RF, better predictions could be achieved by stacking INRF elements in cascade, so that results in one stage are refined in the next. The INRF is also shown to be extremely effective in improving the performance of ANNs in terms of classification error and robustness to adversarial attacks.37: while it is true that goal-driven ANNs are the state of the art in predicting cortical responses, by their own design they can’t yield a model that falls outside the regular L+NL paradigm. While possible in theory, in practice the chances of obtaining the INRF from the hidden layers of an ANN trained for some visual task are virtually zero, as this could only come after designing and training a colossal, unprecedented in size and highly inefficient network architecture.In many common scenarios the INRF reduces to the linear RF, but in general it can’t be expressed efficiently in L+NL form. In our opinion, this has been the key reason why a model like INRF has not been proposed before, despite the wide variety of approaches that over the years have been introduced to model the nonlinear nature of visual phenomena: as all these vision models follow the L+NL paradigm (because they are grounded in the notion of a linear RF), they are not able to implement operations like the ones involved by the INRF unless there is an explosion in the dimension of the linear part. Furthermore, we argue that a model like INRF would most definitely not have appeared if we had followed the current trend of studying vision by using deep ANNsx. For visual perception, we are assessing the ability of our model to predict color appearance phenomena. Another important aspect is that of transmission of information, which goes beyond redundancy reduction  in the response58. Future research should quantify the ability of INRF layers in information transmission from the stimuli into the internal image representation, where the nature of the noise in INRF units (not considered in this work) will be of paramount relevance, following60. And for ANNs with INRF modules, we’re studying other configurations , the mathematical properties of the INRFnet that make it more resistant to adversarial attacks, the potential of an INRFnet to replicate visual illusions better than CNNs61, and INRFnet implementations for input signals other than images.There are many interesting questions that spring up from our proposed model, some of which we’re currently investigating. We’re also interested in analyzing how changes in To conclude, our results suggest a change of paradigm for vision science as well as for artificial intelligence, and open up groundbreaking opportunities for vision research that can potentially bring a game-change in the field, by constituting the new building block with which to produce neural models that perform better in predicting the responses to natural stimuli. The potential effect of fully reaching this goal is immense for vision science, allowing to better understand how the brain processes signals, to help establish and analyze neural circuits, to see in which way neural signals relate to visual perception, and to develop accurate models of image appearance. This in turn would permit to create methods to produce uniform test batteries for easier replication of experiments in neuroimaging and psychophysics, develop better image processing and computer vision algorithms for many different key problems simply by minimizing perceptual error metrics, manufacture cameras and displays that faithfully match our perception, tailor them to individuals suffering from a visual deficiency, etc. The INRF can allow to create ANNs that perform more similarly to a human observer, in the sense that they become robust to adversarial attacks, easier to train, and have better generalization properties. These are all some of the most significant problems currently affecting ANNs, hence the potential impact of our proposal may be quite substantial given the ubiquity of ANNs in research and the industry.I of width 512 points, consisting of alternating black and white segments of width b as background and a superimposed center segment of value u and width 64. In order to simulate retinal blurring, I is convolved with a Gaussian kernel of standard deviation 4, yielding m and g have a width of 85 and a constant value of 1/85, the kernel w has a width of 512 and a constant value of 1/512, O, our model of the cell response, whose center point has some value v. We repeat this process for different values of the center region in the stimulus, varying u from 0 (black) to 1 (white). Each panel in Fig. v as a function of stimulus intensity u, for a given bar width b.The plots in Fig. Stimuli were displayed on a Philips 109B CRT monitor of 1280 by 960 pixels and 75 Hz and a luminance range from 0.65 to 75 cdmSubjects viewed eleven uniform circles with 142: S is calculated using the formula 62, where m and w are Gaussian kernels with standard deviation equal to 52 and 178 pixels, respectively, g is a delta function, The brightness perception curves produced by the INRF model in Fig. m a linear kernel, DoG a kernel defined as a difference of two Gaussian kernels, m is a Gaussian kernel with standard deviation equal to 52, the Gaussian kernels that compose DoG have a standard deviation of 30 and 60 pixels respectively, m is again a Gaussian kernel with standard deviation equal to 52, the standard deviation of the Gaussian kernels of DoG is 40 and 50 pixels respectively, Results shown in Fig. ase Fig. a, m is a52. Pearson correlation with MOS in TID2013 for LPIPS was obtained from54. The INRF-IQ metric for a pair of reference R and distorted D images is calculated as follows. The images are converted from sRGB to CIELAB color space. Note that transform to CIELAB to compute lightness plays the role of the Naka–Rushton photoreceptor nonlinearity used in the other experiments. Their respective lightness channels , and we also compute our model on 25 noisy realizations in order to have stable results. As seen in Fig. 56, where none of the state-of-the-art linear RF vision models that were tried were able to replicate this behaviour with fixed parameters. Our parameters are the same -in terms of visual angles- as in the crispening and image quality tests, and they are the following: kernel m is a Gaussian of standard deviation 54 pixels, kernel w is a Gaussian of standard deviation 183 pixels, g is a Dirac delta, the nonlinearity is The results for this experiment are shown in Fig. m is a Gaussian of standard deviation 100, kernel w is a Gaussian of standard deviation 342, g is a Dirac delta, the nonlinearity is The results in Fig. m, w are simple Gaussians, g is a Dirac delta and the nonlinearity p, q for the positive and negative regions  so as to provide a good fit of the INRF results to the observer data on White’s illusion. This final set of parameter values, which is the one detailed above, was verified to work also for the “crispening” effect and the image quality metric, as well as for the irradiation illusion.First, for the “crispening” effect, the standard deviation of g is just a Delta function and m and w are the same kernel, which is the one to be learned. More details on this formulation can be found in the Supplementary Material The INRF model can be fully integrated within a convolutional neural network (CNN) framework. To do so, we just replace the convolution with linear filters in the convolutional layers by INRF-modules based on the proposed model (Eq. ) and we 63, CIFAR1064, CIFAR10064, and SVHN65 without any preprocessing  or data augmentation strategy  with 3 convolutional layers , followed by a global average pooling with kernel size 8 before a fully-connected output layer. Additionally to activation and pooling layers (disposed after the first two layers), batch normalization layers were added after each convolutional layer.In order to compare the performance of our proposed model with regards to a standard CNN we present a case study on image classification. More in detail, we look at four standard datasets: MNIST68 in their ResNet20 CIFAR10 experiment. We use In MNIST, CIFAR10, and CIFAR100 we use a train-validation split strategy  to find the best parameters and report performance in the test set. For SVHN we prepared the data as seen in previous works: 400 samples per class from the training set and 200 samples per class from extra set are used as validation set, the remaining 598,388 images (mixing up training and extra set) are used for training. We report the error in the test set using the best model in the validation set. We train in all the datasets using Adam optimizer and a 69, DeepFool70, and Carlini & Wagner 71 in the MNIST-trained networks and FGSM and DeepFool for CIFAR10-trained networks. For all the attacks we used their implementations from the Adversarial Robustness Toolbox (ART)72.Results from Tables Supplementary material 1