> For the complete documentation index, see [llms.txt](https://elishai.gitbook.io/neuromorphic-engineering/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://elishai.gitbook.io/neuromorphic-engineering/algorithm-designer-perspective/12.-the-neural-engineeringframework-nef/nef-transformation.md).

# NEF: Transformation

Chapter 12.1.2

Learn about the second principle of NEF: transformation

{% embed url="<https://www.youtube.com/watch?v=zo8I_NtHtWI&list=PLYLu6sY3jnoXTrkfA_WMn0nF_FGz-MaiJ&index=3&t=8s>" %}

{% hint style="info" %}
Read Chapter 12.1.2
{% endhint %}

### Python / Nengo demonstration

Imports:

```
import nengo
from nengo.utils.matplotlib import rasterplot
from nengo.processes import WhiteNoise
import numpy as np
import matplotlib.pyplot as plt
```

### Transforming sin(x) to 2sin(x) by decoder scaling

```
T = 1.0
max_freq = 5

model = nengo.Network()

with model:
    stim = nengo.Node(lambda t: 0.5*np.sin(10*t))
    ensA = nengo.Ensemble(100, dimensions=1)
    ensB = nengo.Ensemble(100, dimensions=1)
    
    nengo.Connection(stim, ensA)
    nengo.Connection(ensA, ensB, transform=2) #function=lambda x: 2*x)
    
    stim_p = nengo.Probe(stim)
    ensA_p = nengo.Probe(ensA, synapse=.01)
    ensB_p = nengo.Probe(ensB, synapse=.01)
    ensA_spikes_p = nengo.Probe(ensA.neurons, 'output')
    ensB_spikes_p = nengo.Probe(ensB.neurons, 'output')
   
sim = nengo.Simulator(model, seed=4)
sim.run(T)

t = sim.trange()
plt.figure(figsize=(6, 4))
plt.ax = plt.gca()
plt.plot(t, sim.data[stim_p],'r', linewidth=4, label='x')
plt.plot(t, sim.data[ensA_p],'g', label='$\hat{x}$')
plt.plot(t, sim.data[ensB_p],'b', label='$f(\hat{x})=2\hat{x}$')
plt.legend()
plt.ylabel("Output")
plt.xlabel("Time")
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5OpGgstdKw3mw0IYI%2FScreen%20Shot%202021-09-08%20at%2020.14.22.png?alt=media\&token=9cd0fb3e-909c-402c-a823-8fbfd905c88c)

### Transforming sin(x) to sin(x)^2 in&#x20;

```
T = 1.0
max_freq = 5

model = nengo.Network()

with model:
    stim = nengo.Node(lambda t: 0.5*np.sin(10*t))
    ensA = nengo.Ensemble(100, dimensions=1)
    ensB = nengo.Ensemble(100, dimensions=1)
    
    nengo.Connection(stim, ensA)
    nengo.Connection(ensA, ensB, function=lambda x: x**2)
    
    stim_p = nengo.Probe(stim)
    ensA_p = nengo.Probe(ensA, synapse=.01)
    ensB_p = nengo.Probe(ensB, synapse=.01)
    ensA_spikes_p = nengo.Probe(ensA.neurons, 'output')
    ensB_spikes_p = nengo.Probe(ensB.neurons, 'output')
   
sim = nengo.Simulator(model, seed=4)
sim.run(T)

t = sim.trange()
plt.figure(figsize=(6, 4))
plt.ax = plt.gca()
plt.plot(t, sim.data[stim_p],'r', linewidth=4, label='x')
plt.plot(t, sim.data[ensA_p],'g', label='$\hat{x}$')
plt.plot(t, sim.data[ensB_p],'b', label='$f(\hat{x})=\hat{x}^2$')
plt.legend()
plt.ylabel("Output")
plt.xlabel("Time")
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5PHSKkwhb4y4R5hmL%2FScreen%20Shot%202021-09-08%20at%2020.16.21.png?alt=media\&token=42905d63-374d-4dc0-8ef2-1749569b4069)

### &#x20;Summation of two shifted sin functions

```
T = 1.0
max_freq = 5

model = nengo.Network()

with model:
    stimA = nengo.Node(lambda t: 0.5*np.sin(10*t))
    stimB = nengo.Node(lambda t: 0.5*np.sin(5*t))
    
    ensA = nengo.Ensemble(100, dimensions=1)
    ensB = nengo.Ensemble(100, dimensions=1)
    ensC = nengo.Ensemble(100, dimensions=1)
    
    nengo.Connection(stimA, ensA)
    nengo.Connection(stimB, ensB)
    nengo.Connection(ensA, ensC)
    nengo.Connection(ensB, ensC)
    
    stimA_p = nengo.Probe(stimA)
    stimB_p = nengo.Probe(stimB)
    ensA_p = nengo.Probe(ensA, synapse=.01)
    ensB_p = nengo.Probe(ensB, synapse=.01)
    ensC_p = nengo.Probe(ensC, synapse=.01)
   
sim = nengo.Simulator(model)
sim.run(T)

t = sim.trange()
plt.figure(figsize=(6,4))
plt.plot(t, sim.data[ensA_p],'b', label="$\hat{x}$")
plt.plot(t, sim.data[ensB_p],'m--', label="$\hat{y}$")
plt.plot(t, sim.data[ensC_p],'k--', label="$\hat{x}+\hat{y}$")
plt.legend(loc='best')
plt.ylabel("Output")
plt.xlabel("Time")
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5PfM33zfeHdFevmoT%2FScreen%20Shot%202021-09-08%20at%2020.18.02.png?alt=media\&token=befddb7e-020e-4286-8aa1-923e77707037)

### Summation of two encoded vectors

```
T = 1
max_freq = 5

model = nengo.Network()

with model:

    stimA = nengo.Node([.3,.5])
    stimB = nengo.Node([.3,-.5])
    
    ensA = nengo.Ensemble(100, dimensions=2)
    ensB = nengo.Ensemble(100, dimensions=2)
    ensC = nengo.Ensemble(100, dimensions=2)
    
    nengo.Connection(stimA, ensA)
    nengo.Connection(stimB, ensB)
    nengo.Connection(ensA, ensC)
    nengo.Connection(ensB, ensC)
    
    stimA_p = nengo.Probe(stimA)
    stimB_p = nengo.Probe(stimB)
    ensA_p = nengo.Probe(ensA, synapse=.02)
    ensB_p = nengo.Probe(ensB, synapse=.02)
    ensC_p = nengo.Probe(ensC, synapse=.02)
   
sim = nengo.Simulator(model)
sim.run(T)

plt.figure()
plt.plot(sim.data[ensA_p][:,0], sim.data[ensA_p][:,1], 'g', label="$\hat{x}$")
plt.plot(sim.data[ensB_p][:,0], sim.data[ensB_p][:,1], 'm', label="$\hat{y}$")
plt.plot(sim.data[ensC_p][:,0], sim.data[ensC_p][:,1], 'k', label="$\hat{x} + \hat{y}$")
plt.ylabel("$x_2$")
plt.xlabel("$x_1$")
plt.legend(loc='best')
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5QCnXSa1VmrkCvCN2%2FScreen%20Shot%202021-09-08%20at%2020.20.25.png?alt=media\&token=96134f98-7544-4052-8248-4b67f0450e27)

### Multiplication of two 2D vectors

```
T = 1.0
max_freq = 5

model = nengo.Network()

with model:
    stimA = nengo.Node(lambda t: 0.5*np.sin(10*t))
    stimB = nengo.Node(lambda t: 0.5*np.sin(5*t))
    
    ensA = nengo.Ensemble(200, dimensions=2)
    ensB = nengo.Ensemble(100, dimensions=1)
    nengo.Connection(stimA, ensA[0])
    nengo.Connection(stimB, ensA[1])

    nengo.Connection(ensA, ensB, function=lambda x: x[0]*x[1])
    
    stimA_p = nengo.Probe(stimA)
    stimB_p = nengo.Probe(stimB)
    ensA_p = nengo.Probe(ensA, synapse=.01)
    ensB_p = nengo.Probe(ensB, synapse=.01)
   
sim = nengo.Simulator(model)
sim.run(T)

t = sim.trange()
plt.figure()
plt.plot(t, sim.data[ensA_p][:,0],'black', label="$\hat{x}[0]$")
plt.plot(t, sim.data[ensA_p][:,1],'black', label="$\hat{x}[1]$")
plt.plot(t, sim.data[ensB_p],'r', label="$\hat{x[0]}\cdot\hat{x[1]}$")
plt.legend(loc='best')
plt.ylabel("Output")
plt.xlabel("Time")
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5Qavl1XpFdGCK-ElW%2FScreen%20Shot%202021-09-08%20at%2020.22.08.png?alt=media\&token=8dfc1ac2-4d59-4121-b0ff-a66353bb8f18)

### Signal gating

```
T = 1.0
max_freq = 5

model = nengo.Network()

with model:
    stimA = nengo.Node(lambda t: 0.5*np.sin(10*t))
    stimB = nengo.Node(lambda t: 0 if (t<.5) else 1)
    
    ensA = nengo.Ensemble(300, dimensions=2, radius=np.sqrt(2))
    ensB = nengo.Ensemble(100, dimensions=1)
    nengo.Connection(stimA, ensA[0])
    nengo.Connection(stimB, ensA[1])
    nengo.Connection(ensA, ensB, function=lambda x: x[0]*x[1])
    
    stimA_p = nengo.Probe(stimA)
    stimB_p = nengo.Probe(stimB)
    ensA_p = nengo.Probe(ensA, synapse=.01)
    ensB_p = nengo.Probe(ensB, synapse=.01)
   
sim = nengo.Simulator(model)
sim.run(T)

t = sim.trange()
plt.figure()
plt.plot(t, sim.data[ensA_p][:,0],'black', label="$\hat{x}[0]$")
plt.plot(t, sim.data[ensA_p][:,1],'blue', label="$\hat{x}[1]$")
plt.plot(t, sim.data[ensB_p],'r', label="$\hat{x[0]}\cdot\hat{x[1]}$")
plt.legend(loc='best')
plt.ylabel("Output")
plt.xlabel("Time")
plt.show()
```

Result:

![](https://1487620879-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MiS7NytljlKs-2Yccqw%2F-Mj5MR8_e0evk70mSUBc%2F-Mj5Quworx1Ajv6mENP5%2FScreen%20Shot%202021-09-08%20at%2020.23.29.png?alt=media\&token=92af9686-f199-4d81-ba3a-6d74de6b71df)

###