Endotaxis: A neuromorphic algorithm for mapping, goal-learning, navigation, and patrolling

Reviewer #1 (Public Review):

This paper presents a highly compelling and novel hypothesis for how the brain could generate signals to guide navigation towards remembered goals. Under this hypothesis, which the authors call "Endotaxis", the brain co-opts its ancient ability to navigate up odor gradients (chemotaxis) by generating a "virtual odor" that grows stronger the closer the animal is to a goal location. This idea is compelling from an evolutionary perspective and a mechanistic perspective. The paper is well-written and delightful to read.

The authors develop a detailed model of how the brain may perform "Endotaxis", using a variety of interconnected cell types (point, map, and goal cells) to inform the chemotaxis system. They tested the ability of this model to navigate in several state spaces, representing both physical mazes and abstract cognitive tasks. The Endotaxis model performed reasonably well across different environments and different types of goals.

The authors further tested the model using parameter sweeps and discovered a critical level of network gain, beyond which task performance drops. This critical level approximately matched analytical derivations.

My main concern with this paper is that the analysis of the critical gain value (gamma_c) is incomplete, making the implications of these analyses unclear. There are several different reasonable ways in which the Endotaxis map cell representations might be normalized, which I suspect may lead to different results. Specifically, the recurrent connections between map cells may either be an adjacency matrix, or a normalized transition matrix. In the current submission, the recurrent connections are an un-normalized adjacency matrix. In a previous preprint version of the Endotaxis manuscript, the recurrent connections between the map cells were learned using Oja's rule, which results in a normalized state-transition matrix (see "Appendix 5: Endotaxis model and the successor representation" in "Neural learning rules for generating flexible predictions and computing the successor representation", your reference 17). The authors state "In summary, this sensitivity analysis shows that the optimal parameter set for endotaxis does depend on the environment". Is this statement, and the other conclusions of the sensitivity analysis, still true if the learned recurrent connections are a properly normalized state-transition matrix?

Overall, this paper provides a very compelling model for how neural circuits may have evolved the ability to navigate towards remembered goals, using ancient chemotaxis circuits.

This framework will likely be very important for understanding how the hippocampus (and other memory/navigation-related circuits) interfaces with other processes in the brain, giving rise to memory-guided behavior.

Reviewer #2 (Public Review):

The manuscript presents a computational model of how an organism might learn a map of the structure of its environment and the location of valuable resources through synaptic plasticity, and how this map could subsequently be used for goal-directed navigation.

The model is composed of 'map cells', which learn the structure of the environment in their recurrent connections, and 'goal-cell' which stores the location of valued resources with respect to the map cell population. Each map cell corresponds to a particular location in the environment due to receiving external excitatory input at this location. The synaptic plasticity rule between map cells potentiates synapses when activity above a specified threshold at the pre-synaptic neuron is followed by above-threshold activity at the post-synaptic neuron. The threshold is set such that map neurons are only driven above this plasticity threshold by the external excitatory input, causing synapses to only be potentiated between a pair of map neurons when the organism moves directly between the locations they represent. This causes the weight matrix between the map neurons to learn the adjacency for the graph of locations in the environment, i.e. after learning the synaptic weight matrix matches the environment's adjacency matrix. Recurrent activity in the map neuron population then causes a bump of activity centred on the current location, which drops off exponentially with the diffusion distance on the graph. Each goal cell receives input from the map cells, and also from a 'resource cell' whose activity indicates the presence or absence of a given values resource at the current location. Synaptic plasticity potentiates map-cell to goal-cell synapses in proportion to the activity of the map cells at time points when the resource cell is active. This causes goal cell activity to increase when the activity of the map cell population is similar to the activity where the resource was obtained. The upshot of all this is that after learning the activity of goal cells decreases exponentially with the diffusion distance from the corresponding goal location. The organism can therefore navigate to a given goal by doing gradient ascent on the activity of the corresponding goal cell. The process of evaluating these gradients and using them to select actions is not modelled explicitly, but the authors point to the similarity of this mechanism to chemotaxis (ascending a gradient of odour concentration to reach the odour source), and the widespread capacity for chemotaxis in the animal kingdom, to argue for its biological plausibility.

The ideas are interesting and the presentation in the manuscript is generally clear. The two principle limitations of the manuscript are: i) Many of the ideas that the model implements have been explored in previous work. ii) The mapping of the circuit model onto real biological systems is pretty speculative, particularly with respect to the cerebellum.

Regarding the novelty of the work, the idea of flexibly navigating to goals by descending distance gradients dates back to at least Kaelbling (Learning to achieve goals, IJCAI, 1993), and is closely related to both the successor representation (cited in manuscript) and Linear Markov Decision Processes (LMDPs) (Piray and Daw, 2021, https://doi.org/10.1038/s41467-021-25123-3, Todorov, 2009 https://doi.org/10.1073/pnas.0710743106). The specific proposal of navigating to goals by doing gradient descent on diffusion distances, computed as powers of the adjacency matrix, is explored in Baram et al. 2018 (https://doi.org/10.1101/421461), and the idea that recurrent neural networks whose weights are the adjacency matrix can compute diffusion distances are explored in Fang et al. 2022 (https://doi.org/10.1101/2022.05.18.492543). Similar ideas about route planning using the spread of recurrent activity are also explored in Corneil and Gerstner (2015, cited in manuscript). Further exploration of this space of ideas is no bad thing, but it is important to be clear where prior literature has proposed closely related ideas.

Regarding whether the proposed circuit model might plausibly map onto a real biological system, I will focus on the mammalian brain as I don't know the relevant insect literature. It was not completely clear to me how the authors think their model corresponds to mammalian brain circuits. When they initially discuss brain circuits they point to the cerebellum as a plausible candidate structure (lines 520-546). Though the correspondence between cerebellar and model cell types is not very clearly outlined, my understanding is they propose that cerebellar granule cells are the 'map-cells' and Purkinje cells are the 'goal-cells'. I'm no cerebellum expert, but my understanding is that the granule cells do not have recurrent excitatory connections needed by the map cells. I am also not aware of reports of place-field-like firing in these cell populations that would be predicted by this correspondence. If the authors think the cerebellum is the substrate for the proposed mechanism they should clearly outline the proposed correspondence between cerebellar and model cell types and support the argument with reference to the circuit architecture, firing properties, lesion studies, etc.

The authors also discuss the possibility that the hippocampal formation might implement the proposed model, though confusingly they state 'we do not presume that endotaxis is localized to that structure' (line 564). A correspondence with the hippocampus appears more plausible than the cerebellum, given the spatial tuning properties of hippocampal cells, and the profound effect of lesions on navigation behaviours. When discussing the possible relationship of the model to hippocampal circuits it would be useful to address internally generated sequential activity in the hippocampus. During active navigation, and when animals exhibit vicarious trial and error at decision points, internally generated sequential activity of hippocampal place cells appears to explore different possible routes ahead of the animal (Kay et al. 2020, https://doi.org/10.1016/j.cell.2020.01.014, Reddish 2016, https://doi.org/10.1038/nrn.2015.30). Given the emphasis the model places on sampling possible future locations to evaluate goal-distance gradients, this seems highly relevant. Also, given the strong emphasis the authors place on the relationship of their model to chemotaxis/odour-guided navigation, it would be useful to discuss brain circuits involved in chemotaxis, and whether/how these circuits relate to those involved in goal-directed navigation, and the proposed model.

Finally, it would be useful to clarify two aspects of the behaviour of the proposed algorithm:

1. When discussing the relationship of the model to the successor representation (lines 620-627), the authors emphasise that learning in the model is independent of the policy followed by the agent during learning, while the successor representation is policy dependent. The policy independence of the model is achieved by making the synapses between map cells binary (0 or 1 weight) and setting them to 1 following a single transition between two locations. This makes the model unsuitable for learning the structure of graphs with probabilistic transitions, e.g. it would not behave adaptively in the widely used two-step task (Daw et al. 2011, https://doi.org/10.1016/j.neuron.2011.02.027) as it would fail to differentiate between common and rare transitions. This limitation should be made clear and is particularly relevant to claims that the model can handle cognitive tasks in general. It is also worth noting that there are algorithms that are closely related to the successor representation, but which learn about the structure of the environment independent of the subjects policy, e.g. the work of Kaelbling which learns shortest path distances, and the default representation in the work of Piray and Daw (both referenced above). Both these approaches handle probabilistic transition structures.

2. As the model evaluates distances using powers of adjacency matrix, the resulting distances are diffusion distances not shortest path distances. Though diffusion and shortest path distances are usually closely correlated, they can differ systematically for some graphs (see Baram et al. cited above).

Reviewer #3 (Public Review):

This paper argues that it has developed an algorithm conceptually related to chemotaxis that provides a general mechanism for goal-directed behaviour in a biologically plausible neural form.

The method depends on substantial simplifying assumptions. The simulated animal effectively moves through an environment consisting of discrete locations and can reliably detect when it is in each location. Whenever it moves from one location to an adjacent location, it perfectly learns the connectivity between these two locations (changes the value in an adjacency matrix to 1). This creates a graph of connections that reflects the explored environment. In this graph, the current location gets input activation and this spreads to all connected nodes multiplied by a constant decay (adjusted to the branching number of the graph) so that as the number of connection steps increases the activation decreases. Some locations will be marked as goals through experiencing a resource of a specific identity there, and subsequently will be activated by an amount proportional to their distance in the graph from the current location, i.e., their activation will increase if the agent moves a step closer and decrease if it moves a step further away. Hence by making such exploratory movements, the animal can decide which way to move to obtain a specified goal.

I note here that it was not clear what purpose, other than increasing the effective range of activation, is served by having the goal input weights set based on the activation levels when the goal is obtained. As demonstrated in the homing behaviour, it is sufficient to just have a goal connected to a single location for the mechanism to work (i.e., the activation at that location increases if the animal takes a step closer to it); and as demonstrated by adding a new graph connection, goal activation is immediately altered in an appropriate way to exploit a new shortcut, without the goal weights corresponding to this graph change needing to be relearnt.

Given the abstractions introduced, it is clear that the biological task here has been reduced to the general problem of calculating the shortest path in a graph. That is, no real-world complications such as how to reliably recognise the same location when deciding that a new node should be introduced for a new location, or how to reliably execute movements between locations are addressed. Noise is only introduced as a 1% variability in the goal signal. It is therefore surprising that the main text provides almost no discussion of the conceptual relationship of this work to decades of previous work in calculating the shortest path in graphs, including a wide range of neural- and hardware-based algorithms, many of which have been presented in the context of brain circuits.

The connection to this work is briefly made in appendix A.1, where it is argued that the shortest path distance between two nodes in a directed graph can be calculated from equation 15, which depends only on the adjacency matrix and the decay parameter (provided the latter falls below a given value). It is not clear from the presentation whether this is a novel result. No direct reference is given for the derivation so I assume it is novel. But if this is a previously unknown solution to the general problem it deserves to be much more strongly featured and either way it needs to be appropriately set in the context of previous work.

Once this principle is grasped, the added value of the simulated results is somewhat limited. These show: 1) in practical terms, the spreading signal travels further for a smaller decay but becomes erratic as the decay parameter (map neuron gain) approaches its theoretical upper bound and decreases below noise levels beyond a certain distance. Both follow the theory. 2) that different graph structures can be acquired and used to approach goal locations (not surprising) .3) that simultaneous learning and exploitation of the graph only minimally affects the performance over starting with perfect knowledge of the graph. 4) that the parameters interact in expected ways. It might have been more impactful to explore whether the parameters could be dynamically tuned, based on the overall graph activity.

Perhaps the most biologically interesting aspect of the work is to demonstrate the effectiveness, for flexible behaviour, of keeping separate the latent learning of environmental structure and the association of specific environmental states to goals or values. This contrasts (as the authors discuss) with the standard reinforcement learning approach, for example, that tries to learn the value of states that lead to reward. Examples of flexibility include the homing behaviour (a goal state is learned before any of the map is learned) and the patrolling behaviour (a goal cell that monitors all states for how recently they were visited). It is also interesting to link the mechanism of exploration of neighbouring states to observed scanning behaviours in navigating animals.

The mapping to brain circuits is less convincing. Specifically, for the analogy to the mushroom body, it is not clear what connectivity (in the MB) is supposed to underlie the graph structure which is crucial to the whole concept. Is it assumed that Kenyon cell connections perform the activation spreading function and that these connections are sufficiently adaptable to rapidly learn the adjacency matrix? Is there any evidence for this? As discussed above, the possibility that an algorithm like 'endotaxis' could explain how the rodent place cell system could support trajectory planning has already been explored in previous work so it is not clear what additional insight is gained from the current model.

Reviewer #1 (Public Review):

This paper presents a highly compelling and novel hypothesis for how the brain could generate signals to guide navigation towards remembered goals. Under this hypothesis, which the authors call "Endotaxis", the brain co-opts its ancient ability to navigate up odor gradients (chemotaxis) by generating a "virtual odor" that grows stronger the closer the animal is to a goal location. This idea is compelling from an evolutionary perspective and a mechanistic perspective. The paper is well-written and delightful to read.

The authors develop a detailed model of how the brain may perform "Endotaxis", using a variety of interconnected cell types (point, map, and goal cells) to inform the chemotaxis system. They tested the ability of this model to navigate in several state spaces, representing both physical mazes and abstract cognitive tasks. The Endotaxis model performed reasonably well across different environments and different types of goals.

The authors further tested the model using parameter sweeps and discovered a critical level of network gain, beyond which task performance drops. This critical level approximately matched analytical derivations.

My main concern with this paper is that the analysis of the critical gain value (gamma_c) is incomplete, making the implications of these analyses unclear. There are several different reasonable ways in which the Endotaxis map cell representations might be normalized, which I suspect may lead to different results. Specifically, the recurrent connections between map cells may either be an adjacency matrix, or a normalized transition matrix. In the current submission, the recurrent connections are an un-normalized adjacency matrix. In a previous preprint version of the Endotaxis manuscript, the recurrent connections between the map cells were learned using Oja's rule, which results in a normalized state-transition matrix (see "Appendix 5: Endotaxis model and the successor representation" in "Neural learning rules for generating flexible predictions and computing the successor representation", your reference 17). The authors state "In summary, this sensitivity analysis shows that the optimal parameter set for endotaxis does depend on the environment". Is this statement, and the other conclusions of the sensitivity analysis, still true if the learned recurrent connections are a properly normalized state-transition matrix?

Overall, this paper provides a very compelling model for how neural circuits may have evolved the ability to navigate towards remembered goals, using ancient chemotaxis circuits.

This framework will likely be very important for understanding how the hippocampus (and other memory/navigation-related circuits) interfaces with other processes in the brain, giving rise to memory-guided behavior.

Reviewer #2 (Public Review):

The manuscript presents a computational model of how an organism might learn a map of the structure of its environment and the location of valuable resources through synaptic plasticity, and how this map could subsequently be used for goal-directed navigation.

The model is composed of 'map cells', which learn the structure of the environment in their recurrent connections, and 'goal-cell' which stores the location of valued resources with respect to the map cell population. Each map cell corresponds to a particular location in the environment due to receiving external excitatory input at this location. The synaptic plasticity rule between map cells potentiates synapses when activity above a specified threshold at the pre-synaptic neuron is followed by above-threshold activity at the post-synaptic neuron. The threshold is set such that map neurons are only driven above this plasticity threshold by the external excitatory input, causing synapses to only be potentiated between a pair of map neurons when the organism moves directly between the locations they represent. This causes the weight matrix between the map neurons to learn the adjacency for the graph of locations in the environment, i.e. after learning the synaptic weight matrix matches the environment's adjacency matrix. Recurrent activity in the map neuron population then causes a bump of activity centred on the current location, which drops off exponentially with the diffusion distance on the graph. Each goal cell receives input from the map cells, and also from a 'resource cell' whose activity indicates the presence or absence of a given values resource at the current location. Synaptic plasticity potentiates map-cell to goal-cell synapses in proportion to the activity of the map cells at time points when the resource cell is active. This causes goal cell activity to increase when the activity of the map cell population is similar to the activity where the resource was obtained. The upshot of all this is that after learning the activity of goal cells decreases exponentially with the diffusion distance from the corresponding goal location. The organism can therefore navigate to a given goal by doing gradient ascent on the activity of the corresponding goal cell. The process of evaluating these gradients and using them to select actions is not modelled explicitly, but the authors point to the similarity of this mechanism to chemotaxis (ascending a gradient of odour concentration to reach the odour source), and the widespread capacity for chemotaxis in the animal kingdom, to argue for its biological plausibility.

The ideas are interesting and the presentation in the manuscript is generally clear. The two principle limitations of the manuscript are: i) Many of the ideas that the model implements have been explored in previous work. ii) The mapping of the circuit model onto real biological systems is pretty speculative, particularly with respect to the cerebellum.

Regarding the novelty of the work, the idea of flexibly navigating to goals by descending distance gradients dates back to at least Kaelbling (Learning to achieve goals, IJCAI, 1993), and is closely related to both the successor representation (cited in manuscript) and Linear Markov Decision Processes (LMDPs) (Piray and Daw, 2021, https://doi.org/10.1038/s41467-021-25123-3, Todorov, 2009 https://doi.org/10.1073/pnas.0710743106). The specific proposal of navigating to goals by doing gradient descent on diffusion distances, computed as powers of the adjacency matrix, is explored in Baram et al. 2018 (https://doi.org/10.1101/421461), and the idea that recurrent neural networks whose weights are the adjacency matrix can compute diffusion distances are explored in Fang et al. 2022 (https://doi.org/10.1101/2022.05.18.492543). Similar ideas about route planning using the spread of recurrent activity are also explored in Corneil and Gerstner (2015, cited in manuscript). Further exploration of this space of ideas is no bad thing, but it is important to be clear where prior literature has proposed closely related ideas.

Regarding whether the proposed circuit model might plausibly map onto a real biological system, I will focus on the mammalian brain as I don't know the relevant insect literature. It was not completely clear to me how the authors think their model corresponds to mammalian brain circuits. When they initially discuss brain circuits they point to the cerebellum as a plausible candidate structure (lines 520-546). Though the correspondence between cerebellar and model cell types is not very clearly outlined, my understanding is they propose that cerebellar granule cells are the 'map-cells' and Purkinje cells are the 'goal-cells'. I'm no cerebellum expert, but my understanding is that the granule cells do not have recurrent excitatory connections needed by the map cells. I am also not aware of reports of place-field-like firing in these cell populations that would be predicted by this correspondence. If the authors think the cerebellum is the substrate for the proposed mechanism they should clearly outline the proposed correspondence between cerebellar and model cell types and support the argument with reference to the circuit architecture, firing properties, lesion studies, etc.

The authors also discuss the possibility that the hippocampal formation might implement the proposed model, though confusingly they state 'we do not presume that endotaxis is localized to that structure' (line 564). A correspondence with the hippocampus appears more plausible than the cerebellum, given the spatial tuning properties of hippocampal cells, and the profound effect of lesions on navigation behaviours. When discussing the possible relationship of the model to hippocampal circuits it would be useful to address internally generated sequential activity in the hippocampus. During active navigation, and when animals exhibit vicarious trial and error at decision points, internally generated sequential activity of hippocampal place cells appears to explore different possible routes ahead of the animal (Kay et al. 2020, https://doi.org/10.1016/j.cell.2020.01.014, Reddish 2016, https://doi.org/10.1038/nrn.2015.30). Given the emphasis the model places on sampling possible future locations to evaluate goal-distance gradients, this seems highly relevant. Also, given the strong emphasis the authors place on the relationship of their model to chemotaxis/odour-guided navigation, it would be useful to discuss brain circuits involved in chemotaxis, and whether/how these circuits relate to those involved in goal-directed navigation, and the proposed model.

Finally, it would be useful to clarify two aspects of the behaviour of the proposed algorithm:

1. When discussing the relationship of the model to the successor representation (lines 620-627), the authors emphasise that learning in the model is independent of the policy followed by the agent during learning, while the successor representation is policy dependent. The policy independence of the model is achieved by making the synapses between map cells binary (0 or 1 weight) and setting them to 1 following a single transition between two locations. This makes the model unsuitable for learning the structure of graphs with probabilistic transitions, e.g. it would not behave adaptively in the widely used two-step task (Daw et al. 2011, https://doi.org/10.1016/j.neuron.2011.02.027) as it would fail to differentiate between common and rare transitions. This limitation should be made clear and is particularly relevant to claims that the model can handle cognitive tasks in general. It is also worth noting that there are algorithms that are closely related to the successor representation, but which learn about the structure of the environment independent of the subjects policy, e.g. the work of Kaelbling which learns shortest path distances, and the default representation in the work of Piray and Daw (both referenced above). Both these approaches handle probabilistic transition structures.

2. As the model evaluates distances using powers of adjacency matrix, the resulting distances are diffusion distances not shortest path distances. Though diffusion and shortest path distances are usually closely correlated, they can differ systematically for some graphs (see Baram et al. cited above).

Reviewer #3 (Public Review):

This paper argues that it has developed an algorithm conceptually related to chemotaxis that provides a general mechanism for goal-directed behaviour in a biologically plausible neural form.

The method depends on substantial simplifying assumptions. The simulated animal effectively moves through an environment consisting of discrete locations and can reliably detect when it is in each location. Whenever it moves from one location to an adjacent location, it perfectly learns the connectivity between these two locations (changes the value in an adjacency matrix to 1). This creates a graph of connections that reflects the explored environment. In this graph, the current location gets input activation and this spreads to all connected nodes multiplied by a constant decay (adjusted to the branching number of the graph) so that as the number of connection steps increases the activation decreases. Some locations will be marked as goals through experiencing a resource of a specific identity there, and subsequently will be activated by an amount proportional to their distance in the graph from the current location, i.e., their activation will increase if the agent moves a step closer and decrease if it moves a step further away. Hence by making such exploratory movements, the animal can decide which way to move to obtain a specified goal.

I note here that it was not clear what purpose, other than increasing the effective range of activation, is served by having the goal input weights set based on the activation levels when the goal is obtained. As demonstrated in the homing behaviour, it is sufficient to just have a goal connected to a single location for the mechanism to work (i.e., the activation at that location increases if the animal takes a step closer to it); and as demonstrated by adding a new graph connection, goal activation is immediately altered in an appropriate way to exploit a new shortcut, without the goal weights corresponding to this graph change needing to be relearnt.

Given the abstractions introduced, it is clear that the biological task here has been reduced to the general problem of calculating the shortest path in a graph. That is, no real-world complications such as how to reliably recognise the same location when deciding that a new node should be introduced for a new location, or how to reliably execute movements between locations are addressed. Noise is only introduced as a 1% variability in the goal signal. It is therefore surprising that the main text provides almost no discussion of the conceptual relationship of this work to decades of previous work in calculating the shortest path in graphs, including a wide range of neural- and hardware-based algorithms, many of which have been presented in the context of brain circuits.

The connection to this work is briefly made in appendix A.1, where it is argued that the shortest path distance between two nodes in a directed graph can be calculated from equation 15, which depends only on the adjacency matrix and the decay parameter (provided the latter falls below a given value). It is not clear from the presentation whether this is a novel result. No direct reference is given for the derivation so I assume it is novel. But if this is a previously unknown solution to the general problem it deserves to be much more strongly featured and either way it needs to be appropriately set in the context of previous work.

Once this principle is grasped, the added value of the simulated results is somewhat limited. These show: 1) in practical terms, the spreading signal travels further for a smaller decay but becomes erratic as the decay parameter (map neuron gain) approaches its theoretical upper bound and decreases below noise levels beyond a certain distance. Both follow the theory. 2) that different graph structures can be acquired and used to approach goal locations (not surprising) .3) that simultaneous learning and exploitation of the graph only minimally affects the performance over starting with perfect knowledge of the graph. 4) that the parameters interact in expected ways. It might have been more impactful to explore whether the parameters could be dynamically tuned, based on the overall graph activity.

Perhaps the most biologically interesting aspect of the work is to demonstrate the effectiveness, for flexible behaviour, of keeping separate the latent learning of environmental structure and the association of specific environmental states to goals or values. This contrasts (as the authors discuss) with the standard reinforcement learning approach, for example, that tries to learn the value of states that lead to reward. Examples of flexibility include the homing behaviour (a goal state is learned before any of the map is learned) and the patrolling behaviour (a goal cell that monitors all states for how recently they were visited). It is also interesting to link the mechanism of exploration of neighbouring states to observed scanning behaviours in navigating animals.

The mapping to brain circuits is less convincing. Specifically, for the analogy to the mushroom body, it is not clear what connectivity (in the MB) is supposed to underlie the graph structure which is crucial to the whole concept. Is it assumed that Kenyon cell connections perform the activation spreading function and that these connections are sufficiently adaptable to rapidly learn the adjacency matrix? Is there any evidence for this? As discussed above, the possibility that an algorithm like 'endotaxis' could explain how the rodent place cell system could support trajectory planning has already been explored in previous work so it is not clear what additional insight is gained from the current model.