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<!-- Any section element inside of this container is displayed as a slide -->
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<section data-external="title.html" data-vertical-align-top data-background-color=#B2BA67 ></section>
<section data-markdown data-vertical-align-top data-background-color=#B2BA67><textarea data-template>
<h1> Lecture 10: Brain-Inspired Neural Networks <br/> </h1>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Anatomy of the Neuron</h2>
<img src="images/neuron_drawing.png" />
- Dendrites: act as inputs ports
- Soma: the body of the cell, usually where inputs converge and where action potentials are generated
- Axon: propagates action potentials along to other neurons
- Terminal Boutons (Synapses): act as outputs of the neuron
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Membrane potential</h2>
<img src="images/bear-03-11.png" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Action Potentials and the Axon</h2>
<img src="images/neuron_drawing.png" />
<img src="images/bear-04-02-1.png" />
<p class=pl> Neurons communicate by all-or-none events called Action Potentials, or ``Spikes''</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> "Biological" neuron model: The Leaky Integrate and Fire Neuron. </h2>
<div class=row>
<div class=column>
<ul>
<li/> Membrane Voltage
$$
\begin{split}
U_i(t) = & V_i(t),\\
\tau_{mem}\frac{\mathrm{d}}{\mathrm{d}t} V_i(t) = & - V_i(t) + I_i(t),\\
\end{split}
$$
<li class=fragment /> Output Spike
$$
S_i = \Theta(U_i)
$$
<li class=fragment /> Synaptic Currents
$$
\begin{split}
I_{i}(t) = \sum_{j\in \text{pre}} W_{ij} S_j(t),
\end{split}
$$
</ul>
</div>
<div class=column>
<img src="images/leaky_if.png" />
</div>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> "Biological" neuron model: The Leaky Integrate and Fire Neuron. </h2>
<img src="images/leaky_if.png" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Recurrent Neural Networks and Working Memory </h2>
<div class=row>
<div class=column>
<p>Working Memory:</p>
<ul>
<li /> A type of short-term memory
<li /> Limited in capacity
<li /> Task- and sensory modality-dependent
<li /> Necessary for cognitive control
</ul>
</div>
<div class=column>
<img src="images/brain_wm.png" />
Human brain areas for working
memory of face identity and
location
</div>
</div>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Neural Correlates of Working Memory </h2>
<img src="images/primate_task.png" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Working Memory of Cognitive Control </h2>
<img src="images/12ax.png" class=stretch />
<p class=ref>O’Reilly and Frank, 2006</p>
<p class=pl> How does the brain implement working memory? </p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Recurrent Neural Networks (RNNs) in Neuroscience </h2>
<ul>
<li > Most models hypothesize that short term memory is a process supported by recurrent connections </li>
<li > An RNN is a network in which the output feeds back into the network (A: Feedforward, B: Recurrent) </li>
<img src="images/feedfoward_vs_recurrent.png" class=small />
<li class=fragment > The majority of connections in the brain are recurrent
<div class=row>
<div class=column>
<img src="images/cortical_microcircuit.png" />
<p class=ref> Douglas and Martin, 1989</p>
</div>
<div class=column>
<blockquote>
... physically mapped the synapses on the dendritic trees (...) in layer 4 of the cat primary visual cortex and found that only 5% of the excitatory synapses arose from the lateral geniculate nucleus (LGN)
</blockquote>
<p class=ref> Binzegger et al. 2004</p>
</div>
</div>
</li>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Recurrent Neural Networks (RNNs) in Neuroscience </h2>
<ul>
<li /> Recurrent connectivity can support sustained activity
<div class=row>
<div class=column>
<img src="images/F1.large.jpg" />
</div>
<div class=column>
<img src="images/F2.large.jpg" class=large />
</div>
</div>
<p class=ref> Murray et al. 2017 </p>
<li class=fragment /> In the brain, how do such neural networks learn?
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Types of Synaptic Plasticity in the Brain</h2>
<div class=row>
<div class=column >
<center>Long-Term Plasticity</center>
</div>
<div class=column >
<center>Short-Term Plasticity</center>
</div>
</div>
<div class=row>
<div class=column >
<img src="images/ltp.png" />
</div>
<div class=column >
<img src="images/stp.jpg" />
</div>
<p class=ref>Tsodyks_Markram97_neur-code</p>
</div class=row>
<div class=row>
<div class=column >
<ul>
<li/> Induced over seconds, persistance over >10 hours
<li/> Many mechanisms: Change in number of Receptors, Release Probability, ...
</ul>
</div>
<div class=column >
<ul>
<li/> Induced over fractions of a second
<li/> Recovery over seconds
<li/> Change in probability of vesicle release, ...
</ul>
</div>
</div class=row>
<p class=ref>Feldman09_syna-mech<p>
<p class=ref>Slide modified from Gerstner <i>et al.</i> 2015</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Hebban Learning</h2>
<div class=row>
<div class=column>
<img src="images/gerstner_hebb_rule.png" class=small />
</div>
<div class=column>
<img src="images/hebb_assemblies.jpg" />
</div>
</div>
When an axon of cell $j$ repeatedly or persistently takes part in activating cell $i$, then $j$'s efficiency as one of the cells activating $i$ is increased
<p class=ref>Hebb49_orga-beha</p>
$$
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) = \eta \nu_i \nu_j
$$
<div class=row>
<div class=column>
<ul>
<li/> Plasticity rule operating on local information
<li/> Captures correlations in activity
<li/> Unsupervised
</ul>
</div>
<div class=column>
<blockquote>''Neurons that fire together wire together''</blockquote>
</div>
</div>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Hebb's Cell Assembly</h2>
<img src="images/bear-24-05.png" />
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Generalized Hebbian Learning</h2>
Generalized Hebbian learning: Introduce dependence on pre-synaptic and post-synaptic activities, and the weight itself:
$$
\begin{split}
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) &= F(w_{ij}, \nu_i, \nu_j)\\\\
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) &= a_0(w_{ij}) + a_1^{pre}(w_{ij})\nu_j + a_1^{post}(w_{ij})\nu_i + a_2(w_{ij})\nu_i \nu_j + \dots \\\\
\end{split}
$$
<center>
<table>
<tr>
<td>Pre (Index j)</td>
<td>On</td>
<td>Off</td>
<td>On</td>
<td>Off</td>
</tr>
<tr>
<td>Post (Index i) </td>
<td>On</td>
<td>On</td>
<td>Off</td>
<td>Off</td>
</tr>
<tr class=fragment >
<td>$\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) \propto \nu_i \nu_j$</td><td>+ </td><td> 0 </td><td> 0 </td><td> 0 </td>
</tr>
<tr class=fragment >
<td>$\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) \propto \nu_i \nu_j - c$</td><td>+ </td><td> - </td><td> - </td><td> - </td>
</tr>
<tr class=fragment >
<td> $\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) \propto (\nu_i - c) \nu_j$ </td><td>(+) </td><td> 0 </td><td> - </td><td> 0 </td>
</tr>
<tr class=fragment >
<td>$\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) \propto (\nu_i - \langle \nu_i \rangle)$</td><td>+ </td><td> - </td><td> - </td><td> + </td>
</tr>
</table>
</center>
<p class=ref>Gerstner_Kistler02_spik-neur</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Modulated Hebb rule</h2>
<b>Modulated Hebb rule: Neuromodulators + Hebbian Learning</b>
$$
\begin{split}
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) &= F(w_{ij}, \nu_i, \nu_j, mod(t))\\\\
\end{split}
$$
Example causes of neuromodulation can be rewards, error, attention, novelty.
<b>Examples:</b>
<ul>
<li class=fragment> Reinforcement learning:
$$
\begin{split}
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) \propto Reward(t) \nu_i \nu_j\\
\end{split}
$$
<p class=ref>Florian07_rein-lear</p>
</li>
<li class=fragment > Supervised Learning:
$$
\begin{split}
\frac{\mathrm{d}}{\mathrm{d} t} w_{ij}(t) &= Error_i(t) a_1^{pre}\nu_j\\
\end{split}
$$
</li>
</ul>
<p class=pl> Some modulated synaptic plasticity rules are recently called three factor rules. </p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Spike-Timing Dependent Plasticity</h2>
<img src="images/bi_poo_scholarpedia.jpeg"/>
<p class=ref>Bi_Poo98_syna-modi</p>
<p class=ref>Jesper Sjostrom and Wulfram Gerstner (2010), Scholarpedia, 5(2):1362.</p>
<ul>
<li>$W$: Learning Window</li>
<li>$t_i^n$: $n$th spike time of post-synaptic neuron $i$</li>
<li>$t_j^f$: $f$th spike time of pre-synaptic neuron $i$</li>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Spike-Timing Dependent Plasticity (STDP)</h2>
<img src="images/bi_poo_scholarpedia.jpeg" />
<p class=ref>Gerstner_Kistler02_spik-neur</p>
Spike-Time Dependent Plasticity Rule:
$$
\Delta w_j = \sum_{f=1}^N \sum_{n=1}^N W(t_i^n - t_j^f)
$$
<p class=ref>Jesper Sjostrom and Wulfram Gerstner (2010), Scholarpedia, 5(2):1362.</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>The Concept of Locality</h2>
For computation to occur on a physical substrate, information much be spatially and temporally local.
<img src=images/local_information.png />
<p class=ref>Neftci_etal19_surrgrad</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Back-Propagating Action Potentials</h2>
<b>STDP requires synapses to sense post-synaptic neuron spike times</b>
<div class=row>
<div class=column > Long-term potentiation is regulated by coincidence of postsynaptic APs and EPSPs
<img src=images/Markram97_Fig_1.png ></img>
<p class=ref>Markram_etal97</p>
</div>
<div class=column > Somadendritic Backpropagation of Action Potentials in Cortical Pyramidal Cells
<img src=images/Buzsaki_Kandel98_Fig2.png />
<p class=ref>Buzsaki and Kandel, J. Neurophysiol. 79: 1587--1591, 1998</p>
</div>
</div class=row>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Spike-Timing Dependent Plasticity (STDP) Implementation</h2>
Online implementation of the Spike-Time Dependent Plasticity Rule using pre-synaptic trace $P_j$ and post-synaptic trace $P_i$:
$$
\begin{split}
\tau_+{\mathrm{d} \over \mathrm{d}t}P_j &= -P_j + S^{pre}_j\\\\
\tau_-{\mathrm{d}\over \mathrm{d}t} P_i &= -P_i + S^{post}_i\\\\
{\mathrm{d} \over \mathrm{d}t}w_{ij} &= a_+ P_j(t) S^{post}_i + a_- P_i(t) S^{pre}_j
\end{split}
$$
<ul>
<li>$\delta(t)$: Delta Dirac function (= spike at time $t$)</li>
<li>$a_+$: Amplitude of LTP $a_-$: Amplitude of LTD</li>
<li>$\tau_+$: Temporal window of LTP</li>
<li>$\tau_-$: Temporal window of LTD</li>
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>STDP as Spike-Based Hebbian Learning</h2>
<img src=images/learning_window_Gerstner_Kistler02.png />
If the pre- and post-synaptic neuron spike times are independent:
$$
\langle \frac{\mathrm{d}}{\mathrm{d}t} w_{ij} \rangle \cong \nu_i \nu_j \underbrace{\int W(s) \mathrm{d}s}_{\text{Area under learning window}}
$$
</textarea></section>
<section data-markdown><textarea data-template>
<h2>STDP as Spike-Based Generalized Hebbian Learning</h2>
<p>A more general spike-time dependent plasticity rule</p>
$$
\frac{\mathrm{d}}{\mathrm{d}t} w_j = a_0(w_{ij}) + a_1^{pre}(w_{ij}) S^{pre}_j + a_1^{post} (w_{ij}) S^{post}_i + a_+ P_j(t) S^{post}_i + a_- P_i(t) S^{pre}_j
$$
<p> If spike times are independent, the temporal average of generalized STDP implements the generalized Hebb rule:</p>
$$
\langle \frac{\mathrm{d}}{\mathrm{d}t} w_{ij} \rangle \cong a_0(w_{ij}) + a_1^{pre}(w_{ij}) \nu_j + a_1^{post}(w_{ij})\nu_i + \nu_i \nu_j \int W(s) \mathrm{d}s
$$
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Notes about STDP</h2>
<ul>
<li/> Rate-based models are consistent with STDP
<li/> Spike-time dependence depends on synapse location wrt soma
<li/> The exponential fit of STDP is for computational convenience
<li/> Update in original model is relative
<img src=images/Bi_Poo_Fig_7.png />
<li/> STDP is not derived from computational requirements
</ul>
<p class=pl >STDP is a measurement, not an accurate mechanistic model!</p>
</textarea></section>
<section data-markdown><textarea data-template>
<h2>Normative Models of Synaptic Plasticity</h2>
<ul>
<li/> Rather than building synaptic plasticity from the bottom-up (as in STDP) Normative model strat with a mathematical model, and make hypotheses about how these could be implemented in synapses.
<li/> Machine learning is a common source of inspiration for normative modeling
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Models of RNNs in Neuroscience: Liquid State Machines</h2>
<img src="images/lsm.svg">
<p class=ref>Maass et al. 2002</p>
<ul>
<li/> One of the earlier trainable models of RNNs
<li/> Also known as Reservoir Learning, Extreme Learning Machines
<li/> Only readout connections are trained in a gradient based fashion. Easy to learn, but does not scale well.
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Models of RNNs in Neuroscience: FORCE Learning</h2>
<div class=row>
<div class=column>
<img src="images/susillo1.png" class=large />
</div>
<div class=column>
<img src="images/susillo2.png">
</div>
</div>
<p class=ref>Susillo and Abbott, 2009</p>
<ul>
<li/> Some recurrent connections are trained
<li/> Does not take into account the history of the activities
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Models of RNNs in Neuroscience: Surrogate Gradient Learning</h2>
<div class=row>
<img src="images/sgdecolle.png">
</div>
<p class=ref>Neftci, Mostafa, Zenke, 2019</p>
<ul>
<li/> Models biological neurons as artificial recurrent neural networks and uses approximate gradient-based learning
<li/> Recurrent connections are trained with partial knowledge of the history
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> "Biological" neuron model: The Leaky Integrate and Fire Neuron. </h2>
<div>
$$
\begin{align*}
U^{t+1} & = \beta U^t + (1-\beta) W S^t_{in} - S^t \tag{Membrane Potential}\\
S^t &= \Theta(U^t-1) \tag{Spike & Reset} \\
\beta & = \exp(\frac{t}{\tau_{mem}})\\
\end{align*}
$$
</div>
<img src="images/leaky_if.png" />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/surr_grad_1.svg" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/surr_grad_2.svg" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/surr_grad_3.svg" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/surr_grad_4.svg" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/surr_grad_5.svg" class=stretch />
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Surrogate Gradient Learning </h2>
<img src="images/sg_loss_cartoon.svg" class=large />
<ul>
<li/> With surrogate gradients, we can train any biological neuron dynamics using gradient backpropagation
<li/> By approximating the temporal credit assignment problem, the gradient descent update is compatible with synaptic plasticity dynamics
</ul>
[](https://colab.research.google.com/drive/1-0S0iL0CVh72tXdBrZglZ5RPilcXSwik?usp=sharing)
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Deep Continuous Local Learning (DECOLLE) </h2>
<h3> </h3>
<div class=row>
<img src="images/decolle.png">
</div>
<p class=ref>Kaiser, Mostafa, Neftci, 2019</p>
<ul>
<li/> State-of-the-art learning of spatio-temporal patterns
</ul>
</textarea></section>
<section data-markdown><textarea data-template>
<h2> Why use recurrent neural networks</h2>
<ul>
<li>
<div><div class=column >Few recurrent connections in shallow neural networks can give them similar power to deep neural networks</div><div class=column >
<img src="images/cornet-brainscore.png" class=large />
<p class=ref>Schrimpf et al. 2019</p></div></div></li>
<li class=fragment >Recurrent neural networks are Turing complete, <em> i.e.</em> they can theoretically emulated any computable algorithm</li>
<li class=fragment >We may not have found the right way to train recurrent neural networks yet <p class=ref>Miller and Hardt, International Conference on Learning Representations, 2019</p> </li>
<li class=fragment >The real world is continuous-time, physical computing systems (<em>e.g.</em> biological neurons) operate under real-time constraints.</li>
</ul>
</textarea></section>
</div>
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