welcome back to the class on neural 欢迎回到关于神经的课程
dynamics after is mathematical detours 动态之后就是数学弯路
we are now back to the main flow of the 我们现在回到了主流
argument for this week for we have 我们有本周的争论
reduced the model of hodgkin-huxley from 将霍奇金 - 霍克斯利的模型从“
four coupled differential equations to 四个耦合微分方程到
two differential equations we are left 我们剩下两个微分方程
with two variables one for the voltage 两个变量之一用于电压
you and the second one an auxiliary 你和第二个助手
variable called W we are now going to 名为W的变量， 我们现在要去做
exploit the mathematical structure of 开发的数学结构
this two dimensional system and analyze 这个二维系统和分析
the behavior in the phase plane and now 在相平面和现在的行为
I'll explain what this means so here is 我会解释这是什么意思所以在这里
again our system of equations if we 如果我们再次是我们的方程组
follow the reduction step study 遵循还原步骤研究
explained starting from the 从开始解释
hodgkin-huxley model we arrive at this 霍奇金 - 霍克斯利模型， 我们到达了这一点
kind of situation which I need to 我需要的一种情况
explain say the horizontal axis is the 解释说， 横轴是
voltage you the vertical axis is the 电压你的垂直轴是
auxiliary variable W now this is a 辅助变量W现在是一个
complicated system of equations f of U U的复数方程组f
and W is itself some kind of nonlinear W本身就是一种非线性的
function however a nice way of getting 功能，但一个很好的方式获得
some insights into the dynamics and 一些洞察动力和
insight into the type of functions that 洞悉功能的类型
we have at our hands it's useful to look 我们在我们的手 看起来 是非常有用
at all points which have no change all 在所有没有改变的地方
points that have no change in the 点没有改变的
voltage the set of points with the UD t 用UD t给点电压组
equals zero is called the Juno line 等于零称为朱诺线
suppose the stimulus is here 假设刺激在这里
then the unit line corresponds to all 那么单元行对应于全部
the points there F as a function of U 那里的F作为U的函数
and W is zero and this is what we have W是零，这就是我们所拥有的
here this is the set of all points there 在这里，这是所有点的集合有
the change the u DT is zero now we can 我们现在可以 改变uDT为零
do the same thing for W and look at all 对W做同样的事情并且看看所有事情
that points which have dw/dt evil zero dw / dt邪恶零点
well these other points but the function 以及这些其他的点，但功能
G of U and W is zero and that's for our U和W的 G 是零，这是我们的
hodgkin-huxley model we choose to do my 霍奇金 - 霍克斯利模型， 我们选择做我的
mentions that's this set of points so we 提到了这一点，所以我们
have two curves called the no clients of 有两条曲线叫做无客户端
the system now there are very special 现在的系统有非常特殊的
points where the u DT is zero and DW DT uDT为零的点和DW DT
is here these are the points where the 在这里这些点在哪里
two lines cross these are called the 两条线交叉这些被称为
fixed points you DT and T wtt are both 0 DT和T wtt的固定点都是0
which means there's no change at all 这意味着根本没有改变
once the voltage to you and the 一旦电压给你和
auxiliary variable W sit here they no 辅助变量W坐在这里他们没有
longer change this is the first fixed 更长的变化是第一次修复
point second fixed point third fixed 第二点定点第三点固定
point it turns out that this fixed point 事实证明，这个固定点
is stable so the new clients allow us to 是稳定的， 所以新的客户允许我们
find graphically fixed points of the 找到图形固定的点
dynamics however with node lines we can 但是我们可以通过节点线获得动态
much more we can do much more than just 我们可以做得更多，而不仅仅是更多
finding fixed points and to see that 找到固定点并看到这一点
let's look at a somewhat simplified 让我们看看有点简化
model this is called the Fitzhugh Nagumo 模型这叫做Fitzhugh Nagumo
model 模型
again I have a stimulus which for the 再次我有一个刺激，这是为了
moment I set equal to zero and this 这一刻我设置为等于零
function f of U and W has a u dependence U和W的函数f具有相关性
u minus one-third u cube minus W it's u减去三分之一u立方体减去W
missing here minus W plus R times I 在这里丢失减W加R次I
which for the moment is zero now if we 现在如果我们现在为零
look at the zero point 看看零点
the unit line the UD t evil 0 to the 单位为UD t邪恶0 线
left hand side is zero the current is 左手边是零电流
zero so I can put W to the other side 零，所以我可以把W放到另一边
and I find W is u minus 1/3 u cube and 我发现W是你减去1/3立方体和
this is a function W takes a value of 这是一个函数W取的值
zero if u is zero it also takes a value 零如果你是零它也需要一个值
of zero if u is square root of three and 如果u是三的平方根，则为零
it starts with a slope of 1 so the 它以1的斜率开始
function will more or less look like 功能会多多少少看起来像
this and then it's symmetric so roughly 这个，然后它是如此粗略的对称
it's something like this and then I have 这是这样的，然后我有
my W node line but the W node line is 我的W节点线，但W节点线是
linear he W DT equals here that means W 这里的WDT等于这意味着W
is easier plus b1 times u now let's put 更容易加上你现在放过的b1次
both constants equal to two so if u is 0 两个常数等于2，所以如果你是0
then W takes a value of 2 and moreover 那么W取值2，而且
it has a slope of 2 so roughly my W 它的斜率大约是我的W
nullcline nullcline
the set of points with DW DT equals 0 is DW DT的点集合等于0
a straight line which 一条直线
through to the two lines cross here so 通过这里交叉这两条线
this would be the fixed point these are 这将是这些的固定点
the first steps but now what can we do 第一步，但现在我们可以做什么
with this kind of graph 用这种图表
I promised then we can do more than just 我承诺， 那么我们可以做的不仅仅是
finding the fixed points in fact we can 事实上我们可以找到固定点
look at the flow in this kind of graph 看这种图中的流程
and this is now redrawn the same kind of 现在又重绘了同样的东西
figure the you node line is the red line 图你的节点线是红线
the W node line is the blue line and my 在W节点线是蓝线，我
fixed point is here we know that if the 在这里我们知道固定点如果
system is here then there's no change 系统在这里没有变化
anymore but what can we say 不再， 但我们可以说什么
if the momentary value of u and W is 如果u和W的瞬时值是
here well you know that the voltage will 在这里，你知道电压会
change because all the points where the 因为所有的点都改变了
voltage does not change are on the red 电压不改变是上冲
line and this point is not on a red line 并且这一点不在红线上
you know that W will change because all 你知道W会因为全部而改变
the points where W is constant would lie W恒定的点将在于
on the blue line and this is not the 在蓝线上，这不是
case here so let's consider small time 这里的情况下，让我们考虑一下小时间
step delta-t 步骤δ-t
in a small time step delta-t I expect a 在一小段时间内， 我期望a
change Delta U which is U DT times delta 改变Delta U，即U DT倍数
T I expect a change of W which is e WD t TI预计W的变化即e WD t
times delta t 时间三角洲t
or I can write it as u TT e WD t times 或者我可以写为u TTêWD t次
delta T so I expect that in a small time 三角洲T，所以我期待在很短的时间
step there's a change of U this is the 步骤有一个U的变化这是
speed at which the system moves in the 系统移动的速度
vault in the voltage direction and this 在电压方向和这个拱顶
is the speed at which the W variable 是 W变量的速度
moves so the flow might be like this 移动，所以流量可能是这样的
this would be Delta U and this would be 这将是DeltaU，这将是
Delta W starting from this point u of W 三角洲W的W的这点u开始
he when W now there's something 他现在W的时候有些东西
interesting now happening on the Newell 纽厄尔现在正在发生有趣的事情
clients on the node line I know on the 我知道的节点线上的客户端
unit line I know that the u DT is zero 单位线我知道你的DT是零
but that means that the change in the U 但这意味着美国的变化
direction unity is here Delta U is zero 方向统一在这里Delta U是零
if Delta U is zero then arrows the flow 如果Delta U为零，则指示流量
arrows must be vertical at this point I 此时箭头必须是垂直的
don't know yet whether they would be 不知道他们是否会
vertically upward or vertically downward 垂直向上或垂直向下
similarly on the W align I know that aw 同样在W 对齐我知道那个aw
DT is zero if DW DT is zero then there 如果DW DT为零，那么DT为零
is no change in doubt in the direction 在方向上没有任何疑问
of W possible Delta W is zero and that 的W可能三角洲W 是零和那
means the direction of flow must be 意味着流动的方向必须是
horizontal eat a leftward or rightward 水平向左或向右吃
pointing now how do I know whether the 现在指出我怎么知道是否
arrows point 箭头指向
towards the left or towards the right 向左或向右
well one way to find out is to evaluate 好的一个方法就是评估
our equations at some point for example 例如，我们的方程在某个时刻
I take this point u equal 1 W equals 0 I 我认为这个点等于1W等于0I
insert this into this equation and for 将此插入到该等式中并用于
the model that I discussed before with a 我之前讨论过的模型
set of parameters I discussed before I 设置的II之前讨论的参数
find that in this case at this location 发现在这种情况下在这个位置
the flow would be slightly upward and 流量会略微上升
slightly to the right now as I get 稍微偏右，现在我得到
closer to the red line the upward 向上靠近红线
component remains there is the 组件仍然存在
horizontal component disappears and 水平分量消失和
therefore the flow here must be avert as 因此这里的流量必须避免
I indicated similarly as I move closer 当我靠近时，我表现得很类似
to the blue line to the W node line the 到蓝线到W节点线
vertical component disappears but the 垂直分量消失，但是
horizontal component remains therefore 水平分量因此保持
the flow is towards the right as 流量是朝着正确的
indicated now gwdt has a certain Azir 表示现在gwdt有一定的阿齐尔
here but the UDT has a certain value as 在这里，但UDT具有一定的价值
we get closer to the fixed point this 我们更接近这个固定点
value of d you did he gets smaller and 你做的d值越小越好
smaller because at the fixed point the 小，因为在固定点处的
length of the arrows is exactly 0 now on 现在箭头的长度正好是0
the other side of the fixed point the 固定点的另一边
direction of arrows has changed the same 箭头方向也改变了
is true for the unil klein arrows get 对于unilklein箭是这样
smaller they turn at the fixed point and 小他们把在固定 点
then this is my total diagram now what 那么这就是我现在的总图
does this mean this mean that this means 这是否意味着这意味着这意味着
that if my momentary value of U and W 如果我的U和W的瞬时值
sits here then the system the dynamics 坐在这里然后系统的动态
will move 将移动
this did a momentary value others if I 如果我这样做， 其他人会 做一个瞬间的价值
sit here the damage will move towards 坐在这里的伤害会走向
the left if I sit in between well I 如果我坐在我之间 ，就离开了
cannot say exactly what the angle is 不能确切地说出角度是什么
other know that it's some combination of 其他知道它的一些组合
upward and leftward and so I can write 向上和向左，所以我可以写
down the flow in the regions between the 向下流动的区域之间的区域
node lines approximately I know roughly 我大概知道节点线
the direction of the arrows so what does 箭头的方向等等
that mean suppose I start at some 这意味着假设我从一些开始
arbitrary point like the point here I 任意点就像这里的点I
know at this point the flow is in this 在这一点上知道流是在这个
direction and then I'm here I know at 方向，然后我在这里我知道
this point the flow is again in this 这一点流量再次在此
direction and I'm here I know at this 方向，我在这里我知道这一点
point the flow is vertically upward then 点流量是垂直向上的
I'm here at this point I mean the region 我在这里指的是这个地区
where the flow is upward and towards the 流向上和朝向
left 剩下
I've been draw it like this I'm here I 我一直在这样画它我在这里我
will follow the direction of flows 将遵循流动的方向
they'll pass here horizontally I will 他们会横向通过我会
move down I will move vertically down I 向下移动， 我会垂直向下移动
will follow the flow here and in the end 将遵循这里和最后的流程
I will end up at the fixed point once 我将在固定点结束一次
you have constructed the flow pattern as 你已经构建的流动模式作为
I did with the blue arrows here you can 我可以在这里用蓝色箭头做
basically read off the dynamics by just 基本上只是通过读取动态
following the flow the flow carries you 随着流量的流动带你
along it's like throwing a little wooden 就像扔一点木头一样
stick into a lively river it's taken 坚持到它被采取的一条热闹的河
along by the flow until at some point it 一直流到它的某个点
may stop there might be a fixed point 可能会停在那里可能有一个固定点
where it gets stuck 卡住的地方
now this is simplified scenario here 现在这是简化的情况
hand-drawn you can think of other 手绘你可以想到其他
situations like this one here again a 像这样的情况再次出现在这里
Fitzhugh Nagumo model there's a slightly Fitzhugh Nagumo模型有一点点
different set of parameters so that we 不同的一组参数让我们
have fixed points let's start at some 有一些固定点让我们开始
arbitrary initial conditions say here I 任意初始条件在这里说我
follow the flow I follow the flow and I 遵循我遵循流程和流程的流程
would end up here let's start with an 最终会在这里让我们从一开始
initial condition here I follow the flow 我在这里遵循流程的初始条件
I follow the flow I go here vertically 我遵循我垂直走向这里的流程
downward I follow the flow I end up here 向下我按照我最终在这里的流程
so from any initial condition I can 所以从任何初始状态我都可以
construct flow patterns which will end 构建将结束的流程模式
at one of the fixed points and in this 在其中一个固定点并在此
case we would have two stable fixed 我们会有两个稳定的固定
points here in the air this fixed point 在这里指出这个固定点在空中
is unstable the flow is away from the 不稳定的流动距离
fixed point let me summarize this we 定点让我总结一下我们
made all this effort for reducing the 为减少这一切而做出了这些努力
hodgkin-huxley model from four equations 霍奇金 - 霍克斯利模型由四个方程组成
to two equations and yes it was worth 到两个等式，是的，这是值得的
the effort because once we are down in 因为一旦我们陷入困境，我们的努力
two dimensions we can analyze the 我们可以分析两个维度
dynamics graphically in this lecture 动态图形在这个讲座
I've introduced the idea of flow 我已经介绍了流程的概念
patterns I have introduced the idea of 我已经介绍了这种想法
milk lines and in the following lectures 奶线和以下讲座
we will exploit these very ideas you 我们会利用这些你的想法
really get a deeper understanding of why 真正深刻理解为什么
near and spikes when you're on spike how 当你在秒杀时如何接近和尖峰
neurons spike we will discuss thresholds 神经元秒杀我们将讨论阈值
people discuss type 1 type 2 behavior 人们讨论1型2型行为
everything will become transparent in 一切都将变得透明
the phase plane during the discussion of 讨论期间的相平面
the dynamics in two dimensions please 请在两个方面的动态
take a moment and answer the quiz 花点时间回答测验
questions before you go on to the next 在你继续下一步之前的问题
lecture 演讲