Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells Dehong Xu1

2025-04-27 0 0 8.67MB 14 页 10玖币

侵权投诉

Conformal Isometry of Lie Group Representation in

Recurrent Network of Grid Cells

Dehong Xu1∗

xudehong1996@ucla.edu

Ruiqi Gao2∗

ruiqig@google.com

Wen-Hao Zhang3

Wenhao.Zhang@UTSouthwestern.edu

Xue-Xin Wei4

weixx@utexas.edu

Ying Nian Wu1

ywu@stat.ucla.edu

1Department of Statistics, UCLA 2Google Research, Brain Team

3Lyda Hill Department of Bioinformatics and O’Donell Brain Institute, UT Southwestern Medical Center

4Departments of Neuroscience and Psychology, UT Austin

Abstract

The activity of the grid cell population in the medial entorhinal cortex (MEC)

of the mammalian brain forms a vector representation of the self-position of the

animal. Recurrent neural networks have been proposed to explain the properties

of the grid cells by updating the neural activity vector based on the velocity

input of the animal. In doing so, the grid cell system effectively performs path

integration. In this paper, we investigate the algebraic, geometric, and topological

properties of grid cells using recurrent network models. Algebraically, we study

the Lie group and Lie algebra of the recurrent transformation as a representation

of self-motion. Geometrically, we study the conformal isometry of the Lie group

representation where the local displacement of the activity vector in the neural

space is proportional to the local displacement of the agent in the 2D physical

space. Topologically, the compact abelian Lie group representation automatically

leads to the torus topology commonly assumed and observed in neuroscience. We

then focus on a simple non-linear recurrent model that underlies the continuous

attractor neural networks of grid cells. Our numerical experiments show that

conformal isometry leads to hexagon periodic patterns in the grid cell responses

and our model is capable of accurate path integration. Code is available at

https:

//github.com/DehongXu/grid-cell-rnn.

1 Introduction

Grid cells [

] in the mammalian dorsal medial entorhinal cortex (MEC) exhibit

striking hexagon grid patterns when the agent (e.g., a rodent) navigates in 2D open environments

[

]. It has been hypothesized that grid cell system performs path

integration [

]. That is, the grid cells integrate the self-motion of the

animal over time to keep track of the animal’s own location in space. This can be implemented by

a recurrent neural network that takes the velocity of the self-motion as input, and transforms the

activities of the grid cells based on the velocity inputs. The animal’s self-position can then be decoded

from the activities of the grid cells.

Collectively, the activities of the grid cell population form a vector in the high-dimensional neural

activity space. This provides a representation of the self-position of the agent in space. The

∗Equal contributions.

Preprint. Under review.

arXiv:2210.02684v2 [q-bio.NC] 7 Nov 2022

recurrent network transforms the activity vector based on the movement velocity of the agent, so

that the transformation is a representation of self-motion, when considered from the perspective of

representational learning. The vector and the transformation together form a representation of the 2D

Euclidean group, which is an abelian additive Lie group.

In a recent paper, [

] studied the group representation property and the isotropic scaling or conformal

isometry property for the general transformation model. In the context of linear transformation models,

they connected this property to the hexagon periodic patterns of the grid cell response maps. With the

conformal isometry property of the transformation of the recurrent neural network, the change of the

activity vector in the neural space is proportional to the input velocity of the self-motion in the 2D

physical space. [

] justiﬁed this condition in terms of robustness to errors or noises in the neurons.

Although [

] studied general transformation model theoretically, they focused on a prototype model

of linear recurrent network numerically, which has an explicit algebraic and geometric structure in

the form of a matrix group of rotations.

In this paper, we study conformal isometry in the context of the non-linear recurrent model that

underlies the hand-crafted continuous attractor neural network (CANN) [

]. In particular, we

will focus on the vanilla version of the recurrent network that is linear in the vector representation of

self-position and is additive in the input velocity, followed by an element-wise non-linear rectiﬁcation

(such as ReLU non-linearity). This model has the simplicity that it is additive in input velocity before

rectiﬁcation. We also explore more complex variants for non-linear recurrent networks, such as

the long short-term memory network (LSTM) [

]. Such models have been studied in recent work

[9, 3, 37, 8].

Our numerical experiments show that our conformal isometry condition is able to learn highly

structured multi-scale hexagon grid code, consistent with the properties of experimentally observed

grid cells of rodents. In addition, our learned model is capable of accurate path integration over a

long distance. Our results generalize previous results of linear network models in [

] to an important

class of non-linear neural network models in theoretical neuroscience that are more physiologically

realistic.

The main contributions of our paper are as follows. (1) We investigate the algebraic, geometric, and

topological properties of the transformation models of grid cells. (2) We study a simple non-linear

recurrent network that underlies the hand-crafted continuous attractor networks for grid cells, and our

numerical experiments suggest that conformal isometry is linked to hexagonal periodic patterns of

grid cells.

2 Lie group representation and conformal isometry

2.1 Representations of self-position and self-motion

We start by introducing the basic components of our model.

x= (x1,x2)∈R2

denotes the position

of the agent. Let

∆x= (∆x1,∆x2)

be the input velocity of the self-motion, i.e., displacement of the

agent within a unit time, so that the agent moves from xto x+∆xafter the unit time.

We assume

v(x) = (vi(x),i=1,...,D)

to be the vector representation of self-position

, where

each element

vi(x)

can be interpreted as the activity of a grid cell when the agent is at position

(vi(x),∀x)

corresponds to the response map of grid cell

is the dimensionality of

, i.e., the

number of grid cells. We refer to the space of

as the “neural space”. We normalize

kv(x)k=1

our experiments.

The set

(v(x),x∈R2)

forms a 2D manifold, or an embedding of

, in the

-dimensional neural

space. We will refer to (v(x),x∈R2)as the “coding manifold”.

With self-motion

∆x

, the vector representation

v(x)

is transformed to

v(x+∆x)

by a general

transformation model:

v(x+∆x) = F(v(x),∆x) = F∆x(v(x)),(1)

where by simplifying

F(·,∆x)

F∆x(·)

in notation, we emphasize that the transformation

dependent on

∆x

. While

v(x)

is a representation of

F∆x

is a representation of

∆x

(v(x),∀x)

and

(F∆x(·),∀∆x)

together form a representation of the 2D additive Euclidean group

, which is

an abelian Lie group. Speciﬁcally, we have the following group representation condition for the

transformation model:

Condition 1.

(Algebraic condition on Lie group representation). For any

, we have (1)

F0(v(x)) =

v(x), and (2) F∆x1+∆x2(v(x)) = F∆x2(F∆x1(v(x))) = F∆x1(F∆x2(v(x))) for any ∆x1and ∆x2.

Condition 1(1) requires that the coding manifold

(v(x),∀x)

are ﬁxed points of

with

∆x=0

. If

is further a contraction off the coding manifold, then

(v(x),∀x)

are the attractor points of

Condition 1(2) requires that moving in one step with displacement

∆x1+∆x2

should be the same

as moving in two steps with displacements

∆x1

and

∆x2

respectively. The group representation

condition is the necessary condition for any valid transformation model (Equation (1)) of grid cells.

Group representation is a central theme in modern mathematics and physics [

]. However, most

of the transformations studied in mathematics and physics are linear transformations that form

matrix groups, and the coding manifold

(v(x),∀x)

is often made implicit. [

] focused on matrix

groups, with

F∆x(v(x)) = M(∆x)v(x)

, so that

M(∆x1+∆x2)v(x) = M(∆x1)M(∆x2)v(x) =

M(∆x2)M(∆x1)v(x)

. [

] studied general transformation model theoretically, but then focused

on the linear transformation model in their numerical experiments. Since the transformations in

recurrent neural networks are usually non-linear, we will focus on non-linear transformation models

in this paper.

2.2 Conformal embedding and conformal isometry

For an inﬁnitesimal self-motion

δx

, it is straightforward to derive a ﬁrst-order Taylor expansion of

the transformation model in Equation (1) with respect to δx

v(x+δx) = F0(v(x)) + F0

0(v(x))δx+o(|δx|)

=v(x) + f(v(x))δx+o(|δx|),(2)

where f(v(x)) = ∂F∆x

∂∆x>(v(x)) |∆x=0is a D×2 matrix.

While

(F∆x,∀∆x∈R2)

forms an abelian Lie group, its derivative of

∆x

, i.e.,

, spans its Lie

algebra. Both F∆xand fare transformations acting on the coding manifold (v(x),∀x).

We identify the conformal isometry condition of the Lie group representation as follows:

Condition 2. (Geometric condition on conformal embedding and conformal isometry).

f(v(x))>f(v(x)) = s2I2,∀x,(3)

where

is the 2-dimensional identity matrix. That is, the two column vectors of

f(v(x))

are of

equal norm s, and are orthogonal to each other.

Under the condition above,

v(x+δx)−v(x)≈f(v(x))δx

is conformal to

δx

, i.e., the 2D local

Euclidean space of

(δx)

in the physical space is embedded conformally as another 2D local Euclidean

space

f(v(x))δx

in the neural activity space. We only need to replace the two orthogonal axes for

δxin the 2D physical space by the two column vectors of f(v(x)) in the neural activity space.

An equivalent statement for the above condition is

kv(x+δx)−v(x)k=skδxk+o(kδxk),∀x,δx.(4)

That is, the displacement in the neural space is proportional to the displacement in the 2D physical

space.

Note that since our analysis is local,

may depend on

. If

is a global constant, then the coding

manifold

(v(x),∀x)

has a ﬂat intrinsic geometry (imagining folding a piece of paper without

stretching it).

[

] studied the conformal isometry property in a local polar coordinate system. In our deﬁnition

above, we use 2D cartesian coordinate system, which is more convenient for the non-linear recurrent

model where

∆x

enters the model additively. In the context of linear recurrent networks, [

] linked

the conformal isometry to the hexagon grid patterns of grid cells.

2.3 2D torus, 2D periodicity, and hexagon grid patterns

The 2D torus topology is commonly assumed a priori in the continuous attractor neural networks

(CANN) for grid cells [

]. The torus topology has been recently supported by analyzing

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ConformalIsometryofLieGroupRepresentationinRecurrentNetworkofGridCellsDehongXu1xudehong1996@ucla.eduRuiqiGao2ruiqig@google.comWen-HaoZhang3Wenhao.Zhang@UTSouthwestern.eduXue-XinWei4weixx@utexas.eduYingNianWu1ywu@stat.ucla.edu1DepartmentofStatistics,UCLA2GoogleResearch,BrainTeam3LydaHillDepartmento...

展开>> 收起<<

Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells Dehong Xu1.pdf

共14页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Conformal Isometry of Lie Group Representation in Recurrent Network of Grid Cells Dehong Xu1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: