State-dependent processing in

Spiking Neural Networks

Renato Duarte

03/05/2018

  • What is the goal?

        (why is it necessary?)

  • How can I actually compute this?

        (what is the relevant I/O mapping?)

  • How is the computation realized?

        (how are the mappings realized?)

Marr, D. (1982). MIT Press.

 Defining the problem...

Systematicity

A Taxonomy of Constraints

Fitch W. T. (2014). Phys. Life Rev., 11:329–364.

An improved conceptual framework:

  • Neurobiology poses very hard constraints on the cognitive architecture
  • Cognition and behavior pose critical constraints on neurobiology

 Defining the problem...

Constraints

Fitch W. T. (2014). Phys. Life Rev., 11:329–364.

An improved conceptual framework:

  • Neurobiology
  • Cognition and behavior
  • Evolution - innate prior structure
  • Learning and development - adaptive systems

 Defining the problem...

 Cognitive constraints

Computation is formally constrained manipulation of representations

Edelman, S. (2012).

What might cognition be if not computation?

van Gelder, T. (1995). Journal of Philosophy, 92:345–381.

 Cognitive constraints

Structured sequence processing

Biological Neural Networks

What might cognition be if not computation?

van Gelder, T. (1995). Journal of Philosophy, 92:345–381.

Computation is formally constrained manipulation of representations

Edelman, S. (2012).

Pattern perception and rule learning in a relevant class of dynamical systems

Structured sequence processing

A problem in cognition that is both generic and fundamental

Petersson, K.M.P. (2007). In IeCS 2007:195–199.

 Cognitive constraints

Dehaene S., et al. (2015). Neuron. 88:2–19.

Reber A.S., et al. (1991). Journal of Experimental Psychology. 17:888–896.

Learning occurs without any requirements of awareness of either the process or the product of acquisition

Structured sequence processing

 Cognitive constraints

Temporally integrated actions do occur even among insects, but they do not reach any degree of complexity until the appearance of the cerebral cortex.

Lashley, K. (1950).

Wilson B., et al. (2015). Nature Communications. 6:8901.

Structured sequence processing

 Cognitive constraints

Temporally integrated actions do occur even among insects, but they do not reach any degree of complexity until the appearance of the cerebral cortex.

Lashley, K. (1950).

Wilson B., et al. (2015). Nature Communications. 6:8901.

Petkov C.I., et al. (2012). Frontiers of Evolutionary Neuroscience. 4:1–24.

Encoder

Decoder

Circuit

 Testing functional hypotheses

Applying cognitive constraints

Define the computation and expected performance

 Testing functional hypotheses

Offline / batch:

\mathbf{W}^{\mathrm{out}} = \mathbf{Z} \mathbf{X}^{T} \left( \mathbf{X} \mathbf{X}^{T} + \beta \mathbb{I} \right)^{-1}

Online:

Readouts as metrics

Define the computation and expected performance

 Testing functional hypotheses

\frac{d x_{i}(t)}{dt} = f_{i} \left( \mathbf{x}(t)\right) + g_{i} \left( \mathbf{x}(t)\right) \mathbf{u}(t)

 Testing functional hypotheses

Applying neurobiological constraints

\frac{dw_{ij}(t)}{dt} = M_{ij}(w_{ij}(t),x^{*}_{i}(t), x^{*}_{j}(t))

Symbolic input sequence:

 

 

Random alternation of 3 "words":

 

Binary encoding

Sigmoidal neurons: weighted-sum + nonlinearity

 

Stability ensured by design (ESP)

 

S_t=\sigma_{1}, \sigma_{2}, ..., \sigma_{T}
\sigma_{i} \in \mathcal{A} = \{a, b, d, i, g, u\}
\{ba, dii, guuu\}

Elman, J. (1993). Cognitive Science.

Jaeger, H. et al. (2009).

Next-symbol prediction:

 

1-step prediction

(only for predictable items)

 

t \in \mathbb{Z}_{+}

Context through recurrence

z(t) =S_{t+1}

 Context-dependence and rule learning

Context through recurrence

 Context-dependence and rule learning

 Grammatical induction through plasticity

Acquisition of complex, rule-governed knowledge

z(t) =P(S_{t+1}=\sigma_{i}|S_{t-(n-1)},...,S_{t})

Task specifications

Duarte R, Seriès P, Morrison A (2014). Proceedings of the 36th Annual Conference of the Cognitive Science Society. 2014:427–432.

Generative rule system:

 

 

 

 

 

Symbolic input sequence:

 

 

 

S_t=\sigma_{1}, \sigma_{2}, ..., \sigma_{T}
\sigma_{i} \in \mathcal{A} = \{\#, M, V, T, R, X\}

Next-symbol predicition:

 

 

Ground truth -

t \in \mathbb{Z}_{+}

Reber, A. (1967). Journal of verbal learning and verbal behavior. 6:855–863.

n=3

System specification

Duarte R, Seriès P, Morrison A (2014). Proceedings of the 36th Annual Conference of the Cognitive Science Society. 2014:427–432.

Lazaar et al. (2009). Frontiers in Computational Neuroscience. :427–432.

Excitatory / Inhibitory neurons

 

Binary (McCulloch-Pitts) neurons

 

Heterogeneous parameters

t \in \mathbb{Z}_{+}

 Grammatical induction through plasticity

System specification

Duarte R, Seriès P, Morrison A (2014). Proceedings of the 36th Annual Conference of the Cognitive Science Society. 2014:427–432.

Biologically-inspired unsupervised learning

eSTDP

Synaptic Normalization

iSTDP

Intrinsic Plasticity

Structural Plasticity

Zheng, P. et al. (2013). PLoS Computational Biology. 9:e1002848.

 Grammatical induction through plasticity

Duarte R, Seriès P, Morrison A (2014). Proceedings of the 36th Annual Conference of the Cognitive Science Society. 2014:427–432.

Compact representations

Grammatical learning is only possible with plasticity

 

Prominent role of iSTDP and SN

  • High performance from compact trajectories
  • Inhibitory adaptation is strictily necessary

 Grammatical induction through plasticity

Duarte R, Seriès P, Morrison A (2014). Proceedings of the 36th Annual Conference of the Cognitive Science Society. 2014:427–432.

Rule Learning

Sensitivity to violations of sequential structure

 

Qualitatively comparable to human behavioral performance

 Grammatical induction through plasticity

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

Task specifications

n \in \mathbb{Z}_{+}

Symbolic input sequence:

 

 

Random or repeating

 

Binary encoding

S_n=\sigma_{1}, \sigma_{2}, ..., \sigma_{T}
\sigma_{i} \in \mathcal{A} = \{S_1, S_2, ..., S_k\}

Identity mapping:

 

Stimulus discrimination / representation

 

 

\mathbf{z}[n] = S_{n}
s_{k}(t)=\frac{1}{\sigma_{u}}\left(\hat{u}_{n}[k]\times\delta(t-n\Delta)\right)\ast g

Input transduction:

 

 

 

Thalamic burst mode

- wake-up call

- signal change

 

 

g(s)=exp(-s/\tau_{r})-exp(-s/\tau_{d})
\mathbf{u}[n]

 State-dependence and representational dynamics

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

Spiking neurons: Leaky Integrate-and-Fire neurons

Fixed, homogeneous parameters

Exponential, conductance-based synapses

 

Circuit state -> low-pass filtered spike trains; sampled at stimulus offset

\frac{dV_{i}}{dt} = f(V_{i})+I(t)
F(i)=\{t_{f}|V_{i}(t_{f})\geq V_{\text{thresh}}\}
\tau_f \frac{x_{i}(t)}{dt} = - x_{i}(t)+S_i(t)
\mathbf{x}[n] = \mathbf{x}_{i}(t^{*})

System specification

S_{i}(t) = \sum_{t_{f} \in F(i)} \delta(t-t_f)

 State-dependence and representational dynamics

\mathrm{N}=10000

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

 State-dependence and representational dynamics

System specification

Adaptive synapses (eSTDP + iSTDP)

van Rossum, M. et al. (2000). The Journal of Neuroscience. 20(23):8812:8821.

Vogels, T. et al. (2012). Science. 334(6062):1569-1573.

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

Stimulus representations

 State-dependence and representational dynamics

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

Stimulus representations

 State-dependence and representational dynamics

Rabinovich, M.I., et al. (2008). PLoS Computational Biology. 4:e1000072.

Duarte, R. & Morrison A. (2014). Frontiers in Computational Neuroscience. 8:124.

Network is insensitive to sequence structure:

  • Memory span < stimulus duration
  • Unable to infer relations among sequence items

Structured sequences?

 State-dependence and representational dynamics

Where does processing memory come from?

Hierarchical processing memory

Hasson U. et al. (2015). Trends in Cognitive Sciences. 19:304–313.

Murray J. D. et al. (2014). Nature Neuroscience. 17:1661–1663.

Memory and timescales in the cortex

Synaptic patterning

Most significant source of regional variation in the adult cortex is the DEX of genes related to synaptic components, in an otherwise relatively homogeneous transcriptome...

Hawrylycz M.J. , et al. (2012). Nature. 489:391–399.

Memory and timescales in the cortex

Duarte, R. , Seeholzer, A., Zilles, K. & Morrison A. (2017). Current Opinion in Neurobiology. 43:156-165.

Synaptic patterning

Hawrylycz M.J. , et al. (2012). Nature. 489:391–399.

... so much so that one can recover cortical organization patterns based on these differences

Memory and timescales in the cortex

Duarte, R. , Seeholzer, A., Zilles, K. & Morrison A. (2017). Current Opinion in Neurobiology. 43:156-165.

Duarte, R. , Seeholzer, A., Zilles, K. & Morrison A. (2017). Current Opinion in Neurobiology. 43:156-165.

Synaptic patterning

Receptor "fingerprints" - default molecular organization

Zilles K. et al. (2015). Cortex. 63:79–89.

Memory and timescales in the cortex

Duarte, R. , Seeholzer, A., Zilles, K. & Morrison A. (2017). Current Opinion in Neurobiology. 43:156-165.

Synaptic patterning

Receptor composition constrains temporal tuning properties

(innate)

Memory and timescales in the cortex

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

 Computational role of heterogeneity

Account for realistic synaptic kinetics in point neuron models

Data-driven Layer 2/3 model

Why Layer 2/3?

- Highly recurrent connectivity

- Sparse, asynchronous activity

- Unique I/O relations

- Small neurites

- "Manageable" complexity

- Complete experimental data

Need to "modularize" heterogeneity

Need more realistic models

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

Task specifications

\mathbf{u}(t+\delta t) \sim \mathcal{U}_{[0, 1]}

Information processing capacity

Sub-population of E neurons

Step-wise constant DC

Dambre J. et al. (2012). Scientific Reports. 2:514.

Ability to perform arbitrary functional mappings:

 

Set targets to cover a broad space of  orthonormal basis functions of u

I_{u}(t) = \rho_{u}\mathbf{u}(t)
\mathrm{z}(t) = \mathrm{z}(u^{-k}(t))
y_{\{l\}} = \{z_{1}, ..., z_{L}\}
y_{\{d_{k}\}} = \prod_{k} \mathcal{P}_{d_{k}} (u(t-k)))

Random input process

 

Measure only properties of the system

 Computational role of heterogeneity

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

3 types of neurons (E / I1 / I2):

- Spiking neurons: Adaptive Leaky Integrate-and-Fire neurons

- Complex synaptic kinetics

- No plasticity

\frac{dV_{i}}{dt} = f(V_{i})+I_{ad, i}(t)+I(t)
\mathbf{x}(t) = V(t)

System specifications

\frac{dI_{ad,i}}{dt} = f_v(V_{i})+f_s(S_{i})

Study the impact of heterogeneity in systems components:

- Homogeneous (Hom)

- Neuronal (Neu)

- Synaptic (Syn)

- Structural (Str)

- Heterogeneous (Het)

 Computational role of heterogeneity

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

Temporal tuning and memory

 Computational role of heterogeneity

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

Variations in receptor composition modulate memory capacity

Memory and Intrinsic timescales

 Computational role of heterogeneity

Duarte, R. & Morrison A. (In Revision). PLoS Computational Biology.

  • Neuronal heterogeneity has the largest functional impact
  • Structural heterogeneity boosts the capacity to process highly nonlinear functions

Processing Capacity

 Computational role of heterogeneity

 Summary I

  • Recurrence allows contextual information to reverberate in the current state
    • Context-dependent representations

 

  • Plasticity allows the network to learn the structure of simple (regular) grammars and developing a reliable predictive model
  • Networks become sensitive to sequence violations and string legality
    • Rule-guided learning

 

  • Learning is successful if trajectories are kept constrained and compact - iSTDP
    • Compact (low-dimensional) representations

Symbolic processing in discrete dynamical systems

  • The macroscopic features of intrinsic dynamics constrain population responses to stimuli
    • Representational "precision" is state-dependent

 

  • Plasticity (iSTDP) acts by maintaining sparse, distributed activity patterns (AI)
    • Active decorrelation
    • Sparse, distributed dynamics

 

  • Stimulus-specific responses evolve through bounded, reproducible trajectories
    • Compact representations
    • Sequential metastability

Stability, precision and state-dependence

 Summary II

  • Patterning of the "synaptic machinery" may prime specific circuits to operate on specific timescales
    • Innate / evolutionary constraints
    • Learning, memory, attention, ...

 

  • Structural and electrophysiological diversity have a significant functional impact
    • Complexity and heterogeneity

 

Heterogeneity and memory

 Summary III

 Discussion / Outlook

Neural responses are constrained to a low-dimensional manifold

 

On-manifold perturbations - high behavioral performance

 

Off-manifold perturbations - learning

Learned sequence structure is reflected in the intrinsic dynamics...

Jazayeri, M., et al. (2017). Neuron. 93:1003–1014.

Mazzucato, L. et al. (2016). Frontiers in Systems Neuroscience. 10(11)

...which, in turn, reflects acquired knowledge

Neural responses are constrained to a low-dimensional manifold

 

On-manifold perturbations - high behavioral performance

 

Off-manifold perturbations - learning

Jazayeri, M., et al. (2017). Neuron. 93:1003–1014.

Mazzucato, L. et al. (2016). Frontiers in Systems Neuroscience. 10(11)

 Discussion / Outlook

Determine the appropriate mappings between cognition/behavior and neural states

Jazayeri, M., et al. (2017). Neuron. 93:1003–1014.

Neural responses are constrained to a low-dimensional manifold

 

On-manifold perturbations - high behavioral performance

 

Off-manifold perturbations - learning

 Discussion / Outlook

The "atoms" of neural computation are the functional outcomes of anatomophysiological and biochemical specializations

 

Compositional sequence learning

- sequences-of-sequences

 Discussion / Outlook

Kiebel SJ, et al. (2009). PLoS Computational Biology. 5:e1000464.

Thank you!

Abigail Morrison

Peggy Series

Karl Magnus Petersson

Karl Zilles

Philipp Weidel

Barna Zajzon

Alexander Seeholzer

Susanne Kunkel

Carlos Toledo-Suarez

 

image/svg+xml

Jannis Schuecker

Sandra Diaz

Yury Zaitsev

Claudia Bachmann

(...)