Emergence of a mutual-growth mechanism in networks evolved by social preference based on indirect utility.

Preferential attachment is an important mechanism in the structural evolution of complex networks. However, though resources on a network propagate and have an effect beyond a direct relationship, growth by preferential attachment based on indirectly propagated resources has not been systematically investigated. Here, we propose a mathematical model of an evolving network in which preference is proportional to a utility function reflecting direct utility from directly connected nodes and indirect utility from indirectly connected nodes beyond the directly connected nodes. Our analysis showed that preferential attachment involving indirect utility forms a converged and hierarchical structure, thereby significantly increasing the indirect utility across the entire network. Further, we found that the structures are formed by mutual growth between adjacent nodes, which promotes a scaling exponent of 1.5 between the number of indirect and direct links. Lastly, by examining several real networks, we found evidence of mutual growth, especially in social networks. Our findings demonstrate a growth mechanism emerging in evolving networks with preference for indirect utility, and provide a foundation for systematically investigating the role of preference for indirect utility in the structural and functional evolution of large-scale social networks.

[Intravascular large B-cell lymphoma diagnosed by total hysterectomy].

Intravascular large B-cell lymphoma (IVLBCL) is a rare form of non-Hodgkin B-cell lymphoma which occurs mainly in capillaries and small blood vessels. Successful diagnosis of IVLBCL is challenging since it lacks tumor formation and presents various clinical manifestations. An 82-year-old Asian female patient presented to our emergency department with a history of general fatigue, weight loss, and fever for two weeks. The patient's random skin biopsy was negative, and her bone marrow biopsy revealed hemophagocytic syndrome with no obvious involvement of lymphoma cells. Gallium scintigraphy showed mild uptake in the uterus, pelvis, and spine. The repetitive bone marrow biopsy result and the endometrial cytology/biopsy were negative; however, the pelvic MRI was compatible with lymphoma, revealing lesions in the corpus uteri, pelvis, and vertebral body. After laparoscopic-assisted vaginal total hysterectomy and bilateral salpingo-oophorectomy, the diagnosis of the Asian variant of IVLBCL was made. Although total hysterectomy remains controversial for elderly patients with declining performance status, we could successfully diagnose the condition and initiate the treatment. The patient's general condition improved soon after starting rituximab, cyclophosphamide, doxorubicin, vincristine, and prednisolone regimen on day 26, and she was discharged on day 45.

Thrombopoietin receptor-based protein-protein interaction screening (THROPPIS).

As protein-protein interactions (PPIs) are involved in many cellular events, development of mammalian cytosolic PPI detection systems is important for drug discovery as well as understanding biological phenomena. We have previously reported a c-kit-based PPI screening (KIPPIS) system, in which proteins of interest were fused with a receptor tyrosine kinase c-kit, leading to intracellular PPI-dependent cell growth. However, it has not been investigated whether PPI can be detected using other receptors. In this study, we employed a thrombopoietin receptor, which belongs to the Type I cytokine receptor family, to develop a thrombopoietin receptor-based PPI screening (THROPPIS) system. To improve the sensitivity of THROPPIS, we examined two strategies of (i) localization of the chimeric receptors on the cell membrane, and (ii) addition of a helper module to the chimeric receptors. Intriguingly, the nonlocalized chimeric receptor showed the best performance of THROPPIS. Furthermore, the addition of the helper module dramatically improved the detection sensitivity. In total, 5 peptide-domain interactions were detected successfully, demonstrating the versatility of THROPPIS. In addition, a peptide-domain interaction was detected even when insulin receptor or epidermal growth factor receptor was used as a signaling domain, demonstrating that this PPI detection system can be extended to other receptors.

Effects of Human Synchronous Hand Movements in Eliciting a Sense of Agency and Ownership.

The self is built as an entity independent from the external world using the human ability to experience the senses of agency and ownership. Humans usually experience these senses during movement. Nevertheless, researchers recently reported that another person's synchronous mirror-symmetrical movements elicited both agency and ownership in research participants. However, it is unclear whether this elicitation was caused by the synchronicity or the mirror symmetry of the movements. To address this question, we investigated the effect of interpersonal synchronization on the self-reported sense of agency and ownership in two conditions, using movements with and without mirror symmetry. Participants performed rhythmic hand movements while viewing the experimenter's synchronous or random hand movements, and then reported their perceptions of agency and ownership in a questionnaire. We observed that agency and ownership were significantly elicited by the experimenter's synchronous hand movements in both conditions. The results suggested that the synchronous movements of another person-rather than mirror- or non-mirror-symmetrical movements-appear to elicit the experience of a sense of agency and ownership. The results also suggested that people could experience these senses not only from their own movements but also from another person's synchronous movements.

Gait-Assist Wearable Robot Using Interactive Rhythmic Stimulation to the Upper Limbs.

Many power-assist wearable exoskeletons have been developed to provide walking support and gait rehabilitation for elderly subjects and gait-disorder patients. Most designers have focused on a direct power-assist to the wearer's lower limbs. However, gait is a coordinated rhythmic movement of four limbs controlled intrinsically by central pattern generators, with the upper limbs playing an important role in walking. Maintaining a normal gait can become difficult as a person ages, because of decreases in limb coordination, stride length, and gait speed. It is known that coordination mechanisms can be governed by the principle of mutual entrainment, in which synchronization develops through the interaction between nonlinear phase oscillators in biological systems. This principle led us to hypothesize that interactive rhythmic stimulation to upper-limb movements might compensate for the age-related decline in coordination, thereby improving the gait in the elderly. To investigate this hypothesis, we developed a gait-assist wearable exoskeleton that employs interactive rhythmic stimulation to the upper limbs. In particular, we investigated the effects on spatial (i.e., hip-swing amplitude) and temporal (i.e., hip-swing period) gait parameters by conducting walking experiments with 12 healthy elderly subjects under one control condition and five upper-limb-assist conditions, where the output motor torque was applied at five different upper-limb swing positions. The results showed a statistically significant increase in the mean hip-swing amplitude, with a mean increment of about 7% between the control and upper-limb-assist conditions. They also showed a statistically significant decrease in the mean hip-swing period, with a mean decrement of about 2.3% between the control and one of the upper-limb-assist conditions. Although the increase in the hip-swing amplitude and the decrease in the hip-swing period were both small, the results indicate the possibility that interactive rhythmic stimulation to the upper limbs might have a positive effect on the gait of the elderly.

The Relationship Between Head Motion Synchronization and Empathy in Unidirectional Face-to-Face Communication.

Embodied synchronization is widely observed in human communication, and is considered to be important in generating empathy during face-to-face communication. However, the quantitative relationship between body motion synchronization and degree of empathy is not fully understood. Therefore, we focused on head motion to investigate phase and frequency differences in head motion synchronization in relation to degree of empathy. We specifically conducted a lecture-based experiment using controlled spoken text divided into two parts: high empathy and low empathy. During the lecture, we measured the acceleration of speakers' and listeners' head motions using an accelerometer, and calculated the synchronization between the time-series data from their acceleration norms. The results showed greater head motion synchronization during high empathy. During high empathy, the speakers' head motions began before those of listeners' in the medium (2.5 to 3.5 Hz) and high (4.0 to 5.0 Hz) frequency ranges, whereas the speakers' head motions tended to start later than those of the listeners' in the low (1.0 to 2.0 Hz) and medium (2.5 to 3.5 Hz) frequency ranges. This suggests that the degree of empathy is reflected by a different relationship between the phase and frequency of head motion synchronization during face-to-face communication.

Author Correction: Node property of weighted networks considering connectability to nodes within two degrees of separation.

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has been fixed in the paper.

The Relationship between Robot's Nonverbal Behaviour and Human's Likability Based on Human's Personality.

At current state, although robotics technology has been immensely developed, the uncertainty to completely engage in human-robot interaction is still growing among people. Many current studies then started to concern about human factors that might influence human's likability like human's personality, and found that compatibility between human's and robot's personality (expressions of personality characteristics) can enhance human's likability. However, it is still unclear whether specific means and strategy of robot's nonverbal behaviours enhances likability from human with different personality traits and whether there is a relationship between robot's nonverbal behaviours and human's likability based on human's personality. In this study, we investigated and focused on the interaction via gaze and head nodding behaviours (mutual gaze convergence and head nodding synchrony) between introvert/extravert participants and robot in two communication strategies (Backchanneling and Turn-taking). Our findings reveal that the introvert participants are positively affected by backchanneling in robot's head nodding behaviour, which results in substantial head nodding synchrony whereas the extravert participants are positively influenced by turn-taking in gaze behaviour, which leads to significant mutual gaze convergence. This study demonstrates that there is a relationship between robot's nonverbal behaviour and human's likability based on human's personality.

Node property of weighted networks considering connectability to nodes within two degrees of separation.

Weighted networks have been extensively studied because they can represent various phenomena in which the diversity of edges is essential. To investigate the properties of weighted networks, various centrality measures have been proposed, such as strength, weighted clustering coefficients, and weighted betweenness centrality. In such measures, only direct connections or entire network connectivity from arbitrary nodes have been used to calculate the connectivity of each node. However, in weighted networks composed of autonomous elements such as humans, middle ranges from each node are also considered to be meaningful for characterizing each node's connectability. In this study, we define a new node property in weighted networks to consider connectability to nodes within a range of two degrees of separation, then apply this new centrality to face-to-face human communication networks in corporate organizations. Our results show that the proposed centrality distinguishes inherent communities corresponding to the job types in each organization with a high degree of accuracy. This indicates the possibility that connectability to nodes within two degrees of separation reveals potential trends of weighted networks that are not apparent from conventional measures.

Voluntary movement affects simultaneous perception of auditory and tactile stimuli presented to a non-moving body part.

The simultaneous perception of multimodal sensory information has a crucial role for effective reactions to the external environment. Voluntary movements are known to occasionally affect simultaneous perception of auditory and tactile stimuli presented to the moving body part. However, little is known about spatial limits on the effect of voluntary movements on simultaneous perception, especially when tactile stimuli are presented to a non-moving body part. We examined the effect of voluntary movement on the simultaneous perception of auditory and tactile stimuli presented to the non-moving body part. We considered the possible mechanism using a temporal order judgement task under three experimental conditions: voluntary movement, where participants voluntarily moved their right index finger and judged the temporal order of auditory and tactile stimuli presented to their non-moving left index finger; passive movement; and no movement. During voluntary movement, the auditory stimulus needed to be presented before the tactile stimulus so that they were perceived as occurring simultaneously. This subjective simultaneity differed significantly from the passive movement and no movement conditions. This finding indicates that the effect of voluntary movement on simultaneous perception of auditory and tactile stimuli extends to the non-moving body part.

Effect of Interpersonal Interaction on Festinating Gait Rehabilitation in Patients with Parkinson's Disease.

UNLABELLED: Although human walking gait rhythms are generated by native individual gait dynamics, these gait dynamics change during interactions between humans. A typical phenomenon is synchronization of gait rhythms during cooperative walking. Our previous research revealed that fluctuation characteristics in stride interval of subjects with Parkinson's disease changed from random to 1/f fluctuation as fractal characteristics during cooperative walking with the gait assist system Walk-Mate, which emulates a human interaction using interactive rhythmic cues. Moreover, gait dynamics were relearned through Walk-Mate gait training. However, the system's clinical efficacy was unclear because the previous studies did not focus on specific gait rhythm disorder symptoms. Therefore, this study aimed to evaluate the effect of Walk-Mate on festinating gait among subjects with Parkinson's disease. Three within-subject experimental conditions were used: (1) preinteraction condition, (2) interaction condition, and (3) postinteraction condition. The only difference between conditions was the interactive rhythmic cues generated by Walk-Mate. Because subjects with festinating gait gradually and involuntarily decreased their stride interval, the regression slope of stride interval as an index of severity of preinteraction festinating gait was elevated. The regression slope in the interaction condition was more gradual than during the preinteraction condition, indicating that the interactive rhythmic cues contributed to relieving festinating gait and stabilizing gait dynamics. Moreover, the gradual regression slope was carried over to the postinteraction condition, indicating that subjects with festinating gait have the potential to relearn stable gait dynamics. These results suggest that disordered gait dynamics are clinically restored through interactive rhythmic cues and that Walk-Mate may have the potential to assist therapists in more effective rehabilitation. TRIAL REGISTRATION: UMIN Clinical Trials Registry UMIN000012591.

The simultaneous perception of auditory-tactile stimuli in voluntary movement.

The simultaneous perception of multimodal information in the environment during voluntary movement is very important for effective reactions to the environment. Previous studies have found that voluntary movement affects the simultaneous perception of auditory and tactile stimuli. However, the results of these experiments are not completely consistent, and the differences may be attributable to methodological differences in the previous studies. In this study, we investigated the effect of voluntary movement on the simultaneous perception of auditory and tactile stimuli using a temporal order judgment task with voluntary movement, involuntary movement, and no movement. To eliminate the potential effect of stimulus predictability and the effect of spatial information associated with large-scale movement in the previous studies, we randomized the interval between the start of movement and the first stimulus, and used small-scale movement. As a result, the point of subjective simultaneity (PSS) during voluntary movement shifted from the tactile stimulus being first during involuntary movement or no movement to the auditory stimulus being first. The just noticeable difference (JND), an indicator of temporal resolution, did not differ across the three conditions. These results indicate that voluntary movement itself affects the PSS in auditory-tactile simultaneous perception, but it does not influence the JND. In the discussion of these results, we suggest that simultaneous perception may be affected by the efference copy.

Detection of Nonverbal Synchronization through Phase Difference in Human Communication.

Nonverbal communication is an important factor in human communication, and body movement synchronization in particular is an important part of nonverbal communication. Some researchers have analyzed body movement synchronization by focusing on changes in the amplitude of body movements. However, the definition of "body movement synchronization" is still unclear. From a theoretical viewpoint, phase difference is the most important factor in synchronization analysis. Therefore, there is a need to measure the synchronization of body movements using phase difference. The purpose of this study was to provide a quantitative definition of the phase difference distribution for detecting body movement synchronization in human communication. The phase difference distribution was characterized using four statistical measurements: density, mean phase difference, standard deviation (SD) and kurtosis. To confirm the effectiveness of our definition, we applied it to human communication in which the roles of speaker and listener were defined. Specifically, we examined the difference in the phase difference distribution between two different communication situations: face-to-face communication with visual interaction and remote communication with unidirectional visual perception. Participant pairs performed a task supposing lecture in the face-to-face communication condition and in the remote communication condition via television. Throughout the lecture task, we extracted a set of phase differences from the time-series data of the acceleration norm of head nodding motions between two participants. Statistical analyses of the phase difference distribution revealed the characteristics of head nodding synchronization. Although the mean phase differences in synchronized head nods did not differ significantly between the conditions, there were significant differences in the densities, the SDs and the kurtoses of the phase difference distributions of synchronized head nods. These results show the difference in nonverbal synchronization between different communication types. Our study indicates that the phase difference distribution is useful in detecting nonverbal synchronization in various human communication situations.

Robust patterning of gene expression based on internal coordinate system of cells.

Cell-to-cell communication in multicellular organisms is established through the transmission of various kinds of chemical substances such as proteins. It is well known that gene expression triggered by a chemical substance in individuals has stable spatial patterns despite the individual differences in concentration patterns of the chemical substance. This fact reveals an important property of multicellular organisms called "robustness", which allows the organisms to generate their forms while maintaining proportion. Robustness has been conventionally accounted for by the stability of solutions of dynamical equations that represent a specific interaction network of chemical substances. However, any biological system is composed of autonomous elements. In general, an autonomous element does not merely accept information on the chemical substance from the environment; instead, it accepts the information based on its own criteria for reaction. Therefore, this phenomenon needs to be considered from the viewpoint of cells. Such a viewpoint is expected to allow the consideration of the autonomy of cells in multicellular organisms. This study aims to explain theoretically the robust patterning of gene expression from the viewpoint of cells. For this purpose, we introduced a new operator for transforming a state variable of a chemical substance from an external coordinate system to an internal coordinate system of each cell, which describes the observation of the chemical substance by cells. We then applied this operator to the simplest reaction-diffusion model of the chemical substance to investigate observation effects by cells. Our mathematical analysis of this extended model indicates that the robust patterning of gene expression against individual differences in concentration pattern of the chemical substance can be explained from the viewpoint of cells if there is a regulation field that compensates for the difference between cells seen in the observation results. This result provides a new insight into the investigation of the mechanism of robust patterning in biological systems composed of individual elements.

Relationship between neural rhythm generation disorders and physical disabilities in Parkinson's disease patients' walking.

Walking is generated by the interaction between neural rhythmic and physical activities. In fact, Parkinson's disease (PD), which is an example of disease, causes not only neural rhythm generation disorders but also physical disabilities. However, the relationship between neural rhythm generation disorders and physical disabilities has not been determined. The aim of this study was to identify the mechanism of gait rhythm generation. In former research, neural rhythm generation disorders in PD patients' walking were characterized by stride intervals, which are more variable and fluctuate randomly. The variability and fluctuation property were quantified using the coefficient of variation (CV) and scaling exponent α. Conversely, because walking is a dynamic process, postural reflex disorder (PRD) is considered the best way to estimate physical disabilities in walking. Therefore, we classified the severity of PRD using CV and α. Specifically, PD patients and healthy elderly were classified into three groups: no-PRD, mild-PRD, and obvious-PRD. We compared the contributions of CV and α to the accuracy of this classification. In this study, 45 PD patients and 17 healthy elderly people walked 200 m. The severity of PRD was determined using the modified Hoehn-Yahr scale (mH-Y). People with mH-Y scores of 2.5 and 3 had mild-PRD and obvious-PRD, respectively. As a result, CV differentiated no-PRD from PRD, indicating the correlation between CV and PRD. Considering that PRD is independent of neural rhythm generation, this result suggests the existence of feedback process from physical activities to neural rhythmic activities. Moreover, α differentiated obvious-PRD from mild-PRD. Considering α reflects the intensity of interaction between factors, this result suggests the change of the interaction. Therefore, the interaction between neural rhythmic and physical activities is thought to plays an important role for gait rhythm generation. These characteristics have potential to evaluate the symptoms of PD.

The effect of visual apparent motion on audiovisual simultaneity.

Visual motion information from dynamic environments is important in multisensory temporal perception. However, it is unclear how visual motion information influences the integration of multisensory temporal perceptions. We investigated whether visual apparent motion affects audiovisual temporal perception. Visual apparent motion is a phenomenon in which two flashes presented in sequence in different positions are perceived as continuous motion. Across three experiments, participants performed temporal order judgment (TOJ) tasks. Experiment 1 was a TOJ task conducted in order to assess audiovisual simultaneity during perception of apparent motion. The results showed that the point of subjective simultaneity (PSS) was shifted toward a sound-lead stimulus, and the just noticeable difference (JND) was reduced compared with a normal TOJ task with a single flash. This indicates that visual apparent motion affects audiovisual simultaneity and improves temporal discrimination in audiovisual processing. Experiment 2 was a TOJ task conducted in order to remove the influence of the amount of flash stimulation from Experiment 1. The PSS and JND during perception of apparent motion were almost identical to those in Experiment 1, but differed from those for successive perception when long temporal intervals were included between two flashes without motion. This showed that the result obtained under the apparent motion condition was unaffected by the amount of flash stimulation. Because apparent motion was produced by a constant interval between two flashes, the results may be accounted for by specific prediction. In Experiment 3, we eliminated the influence of prediction by randomizing the intervals between the two flashes. However, the PSS and JND did not differ from those in Experiment 1. It became clear that the results obtained for the perception of visual apparent motion were not attributable to prediction. Our findings suggest that visual apparent motion changes temporal simultaneity perception and improves temporal discrimination in audiovisual processing.

Interpersonal similarity between body movements in face-to-face communication in daily life.

Individuals are embedded in social networks in which they communicate with others in their daily lives. Because smooth face-to-face communication is the key to maintaining these networks, measuring the smoothness of such communication is an important issue. One indicator of smoothness is the similarity of the body movements of the two individuals concerned. A typical example noted in experimental environments is the interpersonal synchronization of body movements such as nods and gestures during smooth face-to-face communication. It should therefore be possible to estimate quantitatively the smoothness of face-to-face communication in social networks through measurement of the synchronization of body movements. However, this is difficult because social networks, which differ from disciplined experimental environments, are open environments for the face-to-face communication between two individuals. In such open environments, their body movements become complicated by various external factors and may follow unstable and nonuniform patterns. Nevertheless, we consider there to be some interaction during face-to-face communication that leads to the interpersonal synchronization of body movements, which can be seen through the interpersonal similarity of body movements. The present study aims to clarify such interaction in terms of body movements during daily face-to-face communication in real organizations of more than 100 people. We analyzed data on the frequency of body movement for each individual during face-to-face communication, as measured by a wearable sensor, and evaluated the degree of interpersonal similarity of body movements between two individuals as their frequency difference. Furthermore, we generated uncorrelated data by resampling the data gathered and compared these two data sets statistically to distinguish the effects of actual face-to-face communication from those of the activities accompanying the communication. Our results confirm an interpersonal similarity of body movements between two individuals in face-to-face communication, for all the organizations studied, and suggest that some body interaction is behind this similarity.

Interactive rhythmic cue facilitates gait relearning in patients with Parkinson's disease.

To develop a method for cooperative human gait training, we investigated whether interactive rhythmic cues could improve the gait performance of Parkinson's disease patients. The interactive rhythmic cues ware generated based on the mutual entrainment between the patient's gait rhythms and the cue rhythms input to the patient while the patient walked. Previously, we found that the dynamic characteristics of stride interval fluctuation in Parkinson's disease patients were improved to a healthy 1/f fluctuation level using interactive rhythmic cues and that this effect was maintained in the short term. However, two problems remained in our previous study. First, it was not clear whether the key factor underpinning the effect was the mutual entrainment between the gait rhythms and the cue rhythms or the rhythmic cue fluctuation itself. Second, it was not clear whether or not the gait restoration was maintained longitudinally and was relearned after repeating the cue-based gait training. Thus, the present study clarified these issues using 32 patients who participated in a four-day experimental program. The patients were assigned randomly to one of four experimental groups with the following rhythmic cues: (a) interactive rhythmic cue, (b) fixed tempo cue, (c) 1/f fluctuating tempo cue, and (d) no cue. It has been reported that the 1/f fluctuation of stride interval in healthy gait is absent in Parkinson's disease patients. Therefore, we used this dynamic characteristic as an evaluation index to analyze gait relearning in the four different conditions. We observed a significant effect in condition (a) that the gait fluctuation of the patients gradually returned to a healthy 1/f fluctuation level, whereas this did not occur in the other conditions. This result suggests that the mutual entrainment can facilitate gait relearning effectively. It is expected that interactive rhythmic cues will be widely applicable in the fields of rehabilitation and assistive technology.

Construction of unnatural heterodimeric receptors based on IL-2 and IL-6 receptor subunits.

Cells have various receptors on their surface for responding to extracellular signals that involve intercellular communication. Although the mechanism of signal transduction by such wild-type receptors has been studied intensively, there has been minimal effort in investigating whether such receptors could signal when unnaturally coupled. In this study, we investigated whether unnatural receptor pairs comprising interleukin-2 (IL-2) and interleukin-6 (IL-6) receptor subunits could transduce a signal through forced dimerization. We replaced the extracellular domain of IL-2R and IL-6R signaling subunits (IL-2Rß, IL-2Rγ, and gp130) with the FK506-binding protein (FKBP) or the FKBP12-rapamycin binding (FRB) domain, the protein pair known to be heterodimerized by rapamycin. When expressed in a hematopoietic cell line, unnatural heterodimers (IL-2Rß-gp130 and IL-2Rγ-gp130 pairs) successfully transduced a signal. While the IL-2Rγ-gp130 pair maximally mimicked gp130 signaling, the IL-2Rß-gp130 pair gave weaker gp130 signaling and no significant induction of IL-2Rß signaling, indicating a high potential of the IL-2Rγ chain in terms of activating the coupled partners. This is the first report demonstrating that heterodimeric combinations of IL-2R and IL-6R subunits are functional for signaling. Further extension of this approach may attain a creative design of artificial receptor pairs that have distinct signaling properties when compared with natural receptors.

Generation model of positional values as cell operation during the development of multicellular organisms.

Many conventional models have used the positional information hypothesis to explain each elementary process of morphogenesis during the development of multicellular organisms. Their models assume that the steady concentration patterns of morphogens formed in an extracellular environment have an important property of positional information, so-called "robustness". However, recent experiments reported that a steady morphogen pattern, the concentration gradient of the Bicoid protein, during early Drosophila embryonic development is not robust for embryo-to-embryo variability. These reports encourage a reconsideration of a long-standing problem in systematic cell differentiation: what is the entity of positional information for cells? And, what is the origin of the robust boundary of gene expression? To address these problems at a cellular level, in this article we pay attention to the re-generative phenomena that show another important property of positional information, "size invariance". In view of regenerative phenomena, we propose a new mathematical model to describe the generation mechanism of a spatial pattern of positional values. In this model, the positional values are defined as the values into which differentiable cells transform a spatial pattern providing positional information. The model is mathematically described as an associative algebra composed of various terms, each of which is the multiplication of some fundamental operators under the assumption that the operators are derived from the remarkable properties of cell differentiation on an amputation surface in regenerative phenomena. We apply this model to the concentration pattern of the Bicoid protein during the anterior-posterior axis formation in Drosophila, and consider the conditions needed to establish the robust boundary of the expression of the hunchback gene.