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In this section, you use objects from everyday life to help pupils develop important geometrical skills, such as recognising, visualising, describing, sorting, naming, classifying and comparing. The materials below are provided for offline use for your convenience and are not tracked. If you wish to save your progress, please go through the online version. For further information, take a look at our frequently asked questions which may give you the support you need. Skip to main content. Curriculum framework An outline of al Primary Curriculum framework Secondary Science Curriculum framework Subject resources The subject resources are divided into six subject About this course 1 hour study 1 Level 1: Introductory Course description.

There is this interesting thing happening here at the table for me between the two of you [laughing and speaking to Rotondi and Kaprow]. You talk about that experience of talking to people outside of your field, whereas my response with art is more emotional. AK: But that might have been the case if I was a woman. Suppose Gertrude Stein was sitting here.

KA: I would be sitting here with my tongue hanging out. AK: Simply because our modes of communication, especially verbal, have become so adept at separating something from its mode of expression. There is something both fascinating and disturbing to me about the way you are defining discourse. It feels high-falutin and it feels very distanced. AK: That was the point that we were making. To me, environment goes from right here, to right here, to all the way out. And power to me is who I am as a human being and how I use my life and my talent and skills.

Eugenia Butler, The Kitchen Table poster , Text and drawing by Eugenia Butler. Photograph by Christian Krieger. AK: We can talk about empowerment or the corruption of power. And I assure you, any of the most beautiful evenings that I can remember have occurred around food. So there was a kind of schizoid quality for me here. Rotondi talks about the unfinished design of the space where they are eating in the art fair as a factor that led to disjointed conversation.

This is interesting to me because the connection between art and artificiality is one that fascinates me. What is the natural territory, the point of beauty that this takes us to? It may not meet the expectations or the intention you set out from the very beginning. But whatever happens is what happens. It might have been different two days ago when I came in here and the smell of rosemary on the walls was fresh.

All of a sudden I was thinking of when I was a kid and my mother would always cook with rosemary. If you took too much in the beginning and the last person ended up with nothing, you got busted by my mother. Which is probably why I became an architect. I mean, it was sort of a subtractive process, which is conceptual. MR: I was just thinking that an ideal collaboration would be for me to be commissioned to design an elementary school or a high school.

EB: I think the mechanism for collaboration is not yet understood. I think it still has to be invented. Because very often what happens with group collaborations is somebody assumes or is assigned leadership and then there is a hierarchy, which is a team, not collaboration. Or, at certain times one can be more important than the other.

But it was a very painful process of learning that had to develop over a period of years. KA: Additionally, with the Harrisons, their work became more public in a sense of working with outside groups. MR: One thing that runs through most of the creative people I know is that they are empathic. If you take that further out, you reach reciprocity, where there is equality between the two, but the hierarchy can change with each person taking their turn.

MR: For me it always goes back to how safe I felt emotionally growing up in a family. Whether I broke my arm, cracked my head, or got lost, somebody was going to come looking for me. In my family it was the opposite experience. If someone was going to come looking for me, it was either to crack my head or break my arm. MR: I should also add that in a large family kissing and biting were both a sign of love simultaneously. It seems to me that your work [to Kaprow] pioneered interactivity in art making through reciprocity and interactivity.

Because you were responding from moment to moment, it seemed to me that there was reciprocity between one thing and every thing else. AK: That was the general sense, to the extent that a system for having that happen was developed. It was essentially providing a very rudimentary game to a group of people who agreed to play it as well as they could.

Which meant, because it was so simple, that they had to provide all the details. There was a lot of implicit responsibility given to the participants and you can imagine how much confusion there was. It was giving a lot of responsibility, which at the time was probably a little silly…I was not thinking about the consequences. EB: There has been a lot of that in last four days. This whole undertaking has been highly collaborative.

Kim Abeles is an artist who crosses disciplines and media to explore and map the urban environment and chronicle broad social issues. Eugenia P. Butler was a Los Angeles-based artist who played a formative but often overlooked role in the Conceptual art movement.

Aside from an eight-year pause in art making, when she moved to South America to raise her daughter and study shamanism, Butler had a prolific career that spanned over forty years. After returning to the States in the early s, Butler resumed studio practice with a focus on physical objects, including drawings, paintings, sculpture, and furniture.

Beginning in with the Kitchen Table talks, Eugenia also developed public dialogues as part of her practice. She was an integral member of the Los Angeles art community and mentored many young artists. Leila Hamidi is an artist and writer living in Los Angeles. She was profoundly influenced by her mentor, Eugenia Butler, who impressed upon her the value of collaboration and a deep curiosity for all disciplines.

Her interests outside of fine arts have lead to positions with Taschen Books and the Los Angeles-based architecture firm Johnston Marklee.

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She is currently working as a project assistant for Pacific Standard Time, a Getty initiative to explore the post-war art history of Los Angeles through a network of over sixty partner museums and institutions that will culminate in a series of citywide exhibits starting October She also was an authority on mail art and Fluxus.

Allan Kaprow is widely considered the father of Happenings. He studied at New York University art at the undergraduate level, philosophy at the graduate and received his MA from Columbia University in art history. These sensory modalities provide different types of information about various aspects of the external world and serve as complementary cues to improve perception in ambiguous conditions.

For instance, while walking, both the visual input optic flow and the vestibular signal body movement convey useful information about heading-direction, and when integrated together, they give a more reliable estimate of heading-direction than either of the sensory modalities could deliver on its own. Indeed, experimental data has shown that the brain does integrate visual and vestibular cues to infer heading-direction and furthermore, the brain does it in an optimal way as predicted by Bayesian inference Fetsch et al. However, multisensory integration is only a part of multisensory information processing.

While it is appropriate to integrate sensory cues from the same stimulus of interest Figure 1A left , sensory cues from different stimuli need to be segregated rather than integrated in order to distinguish and recognize individual stimuli Figure 1A right. In reality, the brain does not know in advance whether the cues are from the same or different objects.

To accomplish the recognition task, we argue that the brain should carry out multisensory integration and segregation concurrently: a group of neurons integrates sensory cues, while the other computes the disparity information between sensory cues. The interplay between the two groups of neurons determines the final choice of integration versus segregation. A Multisensory integration versus segregation.

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Two underlying stimulus features s 1 and s 2 independently generate two noisy cues x 1 and x 2 , respectively. B Information of single cues is lost after integration. Therefore, from the integrated result, the values of single cues are unknown. Consider the example of integrating the visual and vestibular cues to infer heading-direction, and suppose that both cues have equal reliability.

Thus, if only multisensory integration is performed, the brain faces a chicken and egg dilemma in stimulus perception: without integrating cues, it may be unable to recognize stimuli reliably in an ambiguous environment; but once cues are integrated, the information from individual cues is lost. Concurrent multisensory integration and segregation is able to disentangle this dilemma.

The information of individual cues can be recovered by using the preserved disparity information if necessary, instead of re-gathering new inputs from the external world. While there are other brain regions processing unisensory information, concurrent multisensory integration and segregation provides a unified way to achieve: 1 improved stimulus perception if the cues come from the same stimulus of interest; 2 differentiate and recognize stimuli based on individual cues with little time delay if the cues come from different stimuli of interest.

This processing scheme is consistent with an experimental finding which showed that the brain can still sense the difference between cues in multisensory integration Wallace et al. What are the neural substrates for implementing concurrent multisensory integration and segregation? Previous studies investigating the integration of visual and vestibular cues to infer heading-direction found that in each of two brain areas, namely, the dorsal medial superior temporal area MSTd and the ventral intraparietal area VIP , there are two types of neurons with comparable number displaying different multisensory behaviors: congruent and opposite cells Figure 2 Gu et al.

The tuning curves of a congruent cell in response to visual and vestibular cues are similar Figure 2A , whereas the tuning curve of an opposite cell in response to a visual cue is shifted by degrees half of the period compared to that in response to a vestibular cue Figure 2B. Data analysis and modeling studies suggested that congruent neurons are responsible for cue integration Gu et al.

However, the computational role of opposite neurons remains largely unknown. They do not integrate cues as their responses hardly change when a single cue is replaced by two cues with similar directions. Interestingly, however, their responses vary significantly when the disparity between visual and vestibular cues is enlarged Morgan et al. Similar results were found in VIP Chen et al. A—B Tuning curves of a congruent neuron A and an opposite neuron B. C The histogram of neurons according to their difference between preferred visual and vestibular directions.

Congruent and opposite neurons are comparable in numbers. A—B are adapted from Gu et al. In the present study, we explore whether opposite neurons are responsible for cue segregation in multisensory information processing. Experimental findings showed that many, rather than a single, brain areas exhibit multisensory processing behaviors and that these areas are intensively and reciprocally connected with each other Gu et al.

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The architecture of these multisensory areas is consistent with the structure of a decentralized model Zhang et al. The decentralized model successfully reproduces almost all known phenomena observed in the multisensory integration experiments Fetsch et al. Thus, we consider a decentralized multisensory processing model Zhang et al. As a working example, we focus on studying the inference of heading-direction based on visual and vestibular cues. Our model reproduces the tuning properties of opposite neurons, and verifies that opposite neurons encode the disparity information between cues.

Furthermore, we demonstrate that this disparity information, in coordination with the integration result of congruent neurons, enables the neural system to assess the validity of cue integration and to recover the lost information of individual cues if necessary. Our study sheds light on our understanding of how the brain achieves multisensory information processing efficiently. The brain infers stimulus information based on ambiguous sensory cues. We therefore formulate the multisensory processing problem in the framework of probabilistic inference, and as a working example, we focus on studying the inference of heading-direction based on visual and vestibular cues.

To begin with, we introduce the probabilistic model of multisensory integration. In the form of the von Mises distribution, the likelihood function is given by. In the literature, the study of integration and segregation was often formulated as the issue of causal inference Sato et al. In general, the prior of causal inference consists of more than one components, each corresponding to the causal structure describing the relation between the multiple stimuli.

In this study, we consider a single-component integration prior which has been used in several multisensory integration studies Bresciani et al.


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The integration prior is. This prior reflects that the two stimulus features from the same stimulus tend to have similar values. Ernst and Banks, It has been revealed that in the congruent cueing condition, the brain integrates visual and vestibular cues to infer heading-direction in a manner close to Bayesian inference Gu et al. Since the calculations of the two stimuli are exchangeable, hereafter we only present the results for s 1. The posterior of s 1 is calculated through marginalizing the joint posterior in the above equation,.

The above equation indicates that in multisensory integration, the posterior of a stimulus given combined cues is equal to the product of the posteriors given the individual cues. Notably, although x 1 and x 2 are generated independently by s 1 and s 2 since the visual and vestibular signal pathways are separated , x 2 also provides information of s 1 due to the correlation between s 1 and s 2 specified in the prior.

Equation 4 is the result of Bayesian optimal integration in the form of von Mises distributions, and they are the criteria to judge whether optimal cue integration is achieved in the neural system. A link between the Bayesian criteria for von Mises and Gaussian distributions is presented in Appendix 2. Equation 4 indicates that the von Mises distribution of a circular variable can be interpreted as a vector in a two-dimensional space with its mean and concentration representing the angle and length of the vector, respectively Figure 3A.

In this interpretation, the product of two von Mises distributions can be represented by the summation of the corresponding two vectors. Thus, optimal multisensory integration is equivalent to vector summation see Equation 4 , with each vector representing the posterior of the stimulus given each cue the sum of the two green vectors yields the blue vector in Figure 3B. A Two von Mises distributions plotted in the polar coordinate bottom-left and their corresponding geometric representations top-right.

A von Mises distribution can be represented as a vector, with its mean and concentration corresponding to the angle and length of the vector, respectively. B Geometric interpretation of cue integration and the cue disparity information. The posteriors of s 1 given single cues are represented by two vectors green. Cue integration blue is the sum of the two vectors green , and the cue disparity information red is the difference of the two vectors. C—E The mean and concentration of the integration blue and the cue disparity information red as a function of the cue reliability C , cue disparity D , and reliability of prior E.

The above probabilistic model for multisensory integration assumes that sensory cues are originated from the same stimulus. In case they come from different stimuli, the cues need to be segregated, and the neural system needs to infer stimuli based on individual cues. In practice, the brain needs to differentiate these two situations.

In order to achieve reliable multisensory processing, we propose that while integrating sensory cues, the neural system simultaneously extracts the disparity information between cues, so that with this complementary information, the neural system can assess the validity of cue integration. An accompanying consequence of multisensory integration is that the stimulus information associated with individual cues is lost once they are integrated see Figure 1—figure supplement 1.

Hence besides assessing the validity of integration, extracting both congruent and disparity information by simultaneous integration and segregation enables the system to recover the lost information of individual cues if needed. The disparity information of stimulus one obtained from the two cues is defined to be.

This disparity measure was also used to discriminate alternative moving directions in Jazayeri et al.


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The above equation is the criteria to judge whether the disparity information between two cues is encoded in the neural system. Similar to the geometrical interpretation of multisensory integration, multisensory segregation is interpreted as vector subtraction the subtraction between two green vectors yields the red vector in Figure 3B.

This enables us to assess the validity of multisensory integration. If the two vectors associated with the individual cues have a large disparity, the interpretation becomes the opposite Figure 3D. Thus, by comparing the lengths of the summed and subtracted vectors, the neural system can assess whether two cues should be integrated or segregated. Figure 3C and E further describes the integration and segregation behaviors when the model parameters vary.

As shown in Figure 3C , when the likelihoods have weak reliabilities, the network estimate relies more on the prior. Since the prior encourages integration of the two stimuli, the posterior estimate of stimulus one becomes more biased towards cue 2. At the same time, the mean of the disparity information is biased toward the angular difference of the likelihood peaks.

On the other hand, when the likelihoods are strong, the network estimate relies more on the likelihood, and the posterior estimate of stimulus one becomes less biased toward cue 2. A notable difference between von Mises distribution and Gaussian distribution is that the concentration of integration and disparity information changes with cue disparity in von Mises distribution Figure 3D , while they are fixed in Gaussian distribution Ernst, Before introducing the neural circuit model, we first describe intuitively how opposite neurons encode the cue disparity information and the motivation of the proposed network structure.

Ma et al. That is, the neurons combine the responses to the direct cue and the responses to the indirect cue but shifted to opposite direction. This inspires us to consider a network model where the inputs of indirect cue received by opposite neurons are shifted to opposite direction via connections. Below, we present the network model and demonstrate that the opposite neurons emerge from the connectivity and are able to achieve cue segregation. The neural circuit model we consider has the decentralized structure Zhang et al. Each module carries out multisensory processing via cross-talks between modules.

This decentralized architecture achieves integration in a distributed way and is robust to local failure, and it agrees with the experimental findings that neurons in MSTd and VIP both exhibit multisensory responses and that the two areas are abundantly connected with each other Boussaoud et al. Below we only describe the key features of the decentralized network model, and its detailed mathematical description is presented in Materials and methods Equations Each module has two groups of excitatory neurons, congruent blue circles and opposite neurons red circles.

Each group of excitatory neurons are connected recurrently with each other, and they are all connected to an inhibitory neuron pool purple disk to form a continuous attractor neural network. Each module receives a direct cue through feedforward inputs. Between modules, congruent neurons are connected in the congruent manner blue arrows , while opposite neurons are connected in the opposite manner brown lines.

B Connection profiles between neurons. Black line is the recurrent connection pattern between neurons of the same type in the same module. Blue and red lines are the reciprocal connection patterns between congruent and opposite neurons across modules respectively. C The reliability of the network's estimate of a stimulus is encoded in the peak firing rate of the neuronal population. At each module, there exist two groups of excitatory neurons: congruent and opposite neurons blue and red circles in Figure 4A respectively , and they have the same number of neurons, as supported by experiments Figure 2C Chen et al.

Each group of neurons is modeled as a continuous attractor neural network CANN , mimicking the encoding of heading-direction in neural systems Zhang, ; Wu et al. In the model, the recurrent connection strength is not very strong to support persistent activities after switching off external stimuli, because no persistent activity is observed in multisensory areas. Moreover, neuronal responses in the same group are normalized by the total activity of the population Equation 20 , called divisive normalization Carandini and Heeger, , mimicking the effect of a pool of inhibitory neurons purple disks in Figure 4B.


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Each group of neurons has its individual inhibitory neuron pool, and the two pools of inhibitory neurons in the same module share their overall activities Equation 21 , which intends to introduce mutual inhibition between congruent and opposite neurons. Between modules, neurons of the same type are reciprocally connected with each other Figure 4A—B. For congruent neurons, they are connected with each other in the congruent manner Equation 18 and Figure 4B blue line , that is, the more similar their preferred directions are, the stronger the neuronal connection is.

For opposite neurons, they are connected in the opposite manner Equation 19 and Figure 4B red line , that is, the more different their preferred directions are, the stronger the neuronal connection is.

Introduction

We set the connection profile between the opposite neurons to be of the same strength and width as that between the congruent ones comparing Equations 18 and 19 , ensuring that the tuning functions of the opposite neurons have the similar shape as those of the congruent ones, as observed in the experimental data Chen et al. When sensory cues are applied, the neurons combine the feedforward, recurrent, and reciprocal inputs to update their activities Equation 16 , and the multisensory integration and segregation will be accomplished by the reciprocal connections between network modules.

The results are presented below. Simulating the neural circuit model, we first checked the tuning properties of neurons. The simulation results for an example congruent neuron and an example opposite neuron in module 1 responding to single cues are presented in Figure 5. Thus, the tuning properties of congruent and opposite neurons naturally emerge through the network dynamics.

A—B The tuning curves of an example congruent neuron A and an example opposite neuron B in module 1 under three cueing conditions. The two marginal curves around each contour plot are the unimodal tuning curves. G—H The histogram of the differences of neuronal preferred directions with respect to two cues in module 1 G and module 2 H , when the reciprocal connections across network modules contain random components of roughly the same order as the connections.

Other parameters are the same as those in Figure 4. We further checked the responses of neurons to combined cues and found that when there is no disparity between the two cues, the response of a congruent neuron is enhanced compared to the single cue conditions green line in Figure 5A , whereas the response of an opposite neuron is suppressed compared to its response to the direct cue green line in Figure 5B.

These properties agree with the experimental data Gu et al. Following the experimental protocol Morgan et al. These behaviors agree with the experimental observations Morgan et al. We found that by considering the realistic imperfectness of neuronal reciprocal connections e. In a single instance, we used the population vector to read out the stimulus value Equation 23 Georgopoulos et al. The statistics of the bump position sampled from a collection of instances reflects the posterior distribution of the stimulus estimated by the neural population under the given cueing condition.

A Illustration of the population response of congruent neurons in module 1 when both cues are presented. Color indicates firing rate. Right panel is the temporal average firing rates of the neural population during cue presentation, with shaded region indicating the standard deviation SD.

C—E The temporal average population activities of congruent blue and opposite red neurons in module 1 top row and module 2 bottom row under three cueing conditions: only cue 1 is presented C , only cue 2 is presented D , and both cues are simultaneously presented E.

Symbols: network results; lines: theoretical prediction. The theoretical predictions for the estimates of congruent and opposite neurons are obtained by Equations 4 and 7. To validate the hypothesis that congruent and opposite neurons are responsible for cue integration and segregation respectively, we carried out simulations following the protocol in multisensory experiments Fetsch et al.

Similar population activities exist under cue 2 condition Figure 6D. Compared with single cue conditions, the responses of congruent neurons are enhanced comparing Figure 6E with Figure 6C-D , reflecting the increased reliability of the estimate after cue integration. Indeed, the decoded distribution from congruent neurons sharpens in the combined cue condition and moves to a location between cue 1 and cue 2 Figure 6—figure supplement 1 green , which is a typical phenomenon associated with cue integration.

In contrast, with combined cues, the responses of opposite neurons are suppressed compared with those of the direct cue comparing Figure 6E with Figure 6C-D. Certainly, the distribution of cue disparity information decoded from opposite neurons in combined cue condition is wider than that that under the direct cue condition Figure 6—figure supplement 1 purple. To demonstrate that the network implements cue integration and segregation and how the network encodes the probabilistic model Equations 1 and 2 , we changed a parameter at a time, and then compared the decoded results from congruent and opposite neurons with the theoretical predictions.

Figure 6F—I indicates that the network indeed implements optimal integration and segregation. Moreover, comparing the network results with the results of the probabilistic model, we could find the analogy that the input intensity encodes the reliability of the likelihood Equation 1 , comparing Figure 6F with Figure 3C , and the reciprocal connection strength effectively represents the reliability of the integration prior Equation 2 , comparing Figure 6H with Figure 3E , which is consistent with a previous study Zhang et al.

We further systematically changed the network and input parameters over a large parameter region and compare the network results with theoretical predictions.

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Our results indicated that the network model achieves cue integration and segregation robustly over a large range of parameters Figure 6—figure supplement 2 , as long as the connection strengths are not so large that winner-take-all happens in the network model. The above results elucidate that congruent neurons integrate cues, whereas opposite neurons compute the disparity between cues.

Based on these complementary information, the brain can access the validity of cue integration and can also recover the stimulus information associated with single cues lost due to integration. Below, rather than exploring the detailed neural circuit models, we demonstrate that the brain has resources to implement these two operations based on the activities of congruent and opposite neurons. The competition between congruent and opposite neurons can determine whether the brain should integrate or segregate two cues. The brain can judge the validity of integration based on the competition between these two groups of neurons see more remarks in Conclusions and Discussions.