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Learning and Stability

Explore the mechanisms of synaptic plasticity and their role in learning and stability. Learn about the work of Ramón y Cajal in the 19th century and the contributions of James, Hebb, Martin, and Morris in understanding the synaptic efficiency and firing patterns of neurons. Discover concepts such as cell assembly and recall, and dive into the dynamics of Hebbian Plasticity and the long-term stability of synaptic weights. Find out about different mechanisms of stabilization, including sliding threshold, subtractive and multiplicative normalization, and synaptic scaling.

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Learning and Stability

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  1. Learning and Stability

  2. Learning and Memory Ramón y Cajal, 19th century

  3. Learning and Memory Ramón y Cajal, 19th century

  4. Learning and Memory axon dendrite synapse neuron Ramón y Cajal, 19th century

  5. Learning and Memory axon dendrite synapse neuron Ramón y Cajal, 19th century

  6. Learning and Memory Environmental Input active neuron synapse neuron James, 1890; Hebb, 1949; Martin et al., 2000; Martin and Morris, 2002; …

  7. Learning and Memory i Synaptic Plasticity: If the neurons i and j are active together the synaptic efficiency wij increases. Environmental Input wij j Thus, the influence of neuron j on the firing of neuron i is enhanced. synapse neuron James, 1890; Hebb, 1949; Martin et al., 2000; Martin and Morris, 2002; …

  8. Learning and Memory Synaptic Plasticity: If the neurons i and j are active together the synaptic efficiency wij increases. Recall Cell Assembly: A group of strongly connected neurons which tend to fire together can represent a memory item. A later presented (partial) recall signal of the original learned input has to activate the same group of neurons. James, 1890; Hebb, 1949; Martin et al., 2000; Martin and Morris, 2002; …

  9. Learning and Memory Synaptic Plasticity: If the neurons i and j are active together the synaptic efficiency wij increases. Cell Assembly: A group of strongly connected neurons which tend to fire together can represent a memory item. Synaptic Plasticityand Memory Hypothesis Synaptic Plasticity = Learning James, 1890; Hebb, 1949; Martin et al., 2000; Martin and Morris, 2002; …

  10. Synaptic Plasticity: Hebbian Neuron A Neuron B Long-Term Potentiation (LTP) Synapse u v ω u: input v: output ω: weight μ : learning rate Classical Hebbian plasticity: ω=μ u v What are the long-term dynamics of Hebbian Plasticity?

  11. Learning and Stability

  12. Synaptic Plasticity: Hebbian Neuron A Neuron B Long-Term Potentiation (LTP) Synapse u v ω linear neuron model: v=u ω u: input v: output ω: weight μ : learning rate Classical Hebbian plasticity: ω=μ u v stability analysis The weights follow divergent dynamics

  13. Mechanisms of stabilization dw = m vu (v - Q) m << 1 Sliding threshold: dt dQ = n (v2 - Q) n< 1 dt Subtractive normalization: dw = m (vu – a v2w), a>0 dt Multiplicative normalization:

  14. Sliding Threshold BCM- Rule dw = m vu (v - Q) m << 1 dt As such this rule is again unstable, but BCM introduces a sliding threshold dQ = n (v2 - Q) n < 1 dt Note the rate of threshold change n should be faster than then weight changes (m), but slower than the presentation of the individual input patterns. This way the weight growth will be over-dampened relative to the (weight – induced) activity increase.

  15. Sliding Threshold The sliding threshold directly changes synaptic plasticity by the self-organized adaptation of the plasticity-parameters. , if BCM rule: Sliding Threshold : Increase activity : Decrease activity : LTP LTD Abraham and Bear, 1996; Abraham, 2008

  16. Sliding Threshold light-deprived (less activity) after two days control condition (more activity) Kirkwood et al., 1996 The threshold shifts slowly according to neuronal activity (sensory stimulation).

  17. Sliding Threshold BCM rule: Sliding Threshold : oscillations are stronger than convergence oscillations Yger and Gilson, 2015

  18. Mechanisms of stabilization dw = m vu (v - Q) m << 1 Sliding threshold: biologically unrealistic dt dQ = n (v2 - Q) n< 1 dt Subtractive normalization: dw = m (vu – a v2w), a>0 dt Multiplicative normalization:

  19. Subtractive normalization Subtractive: v <u> With N number of inputs and n a unit vector (all “1”). This yields that n.u is just the sum over all inputs. This normalization is rigidly apply at each learning step. It requires global information (info about ALL inputs), which is biologicallyunrealistic.

  20. Mechanisms of stabilization dw = m vu (v - Q) m << 1 Sliding threshold: biologically unrealistic dt dQ = n (v2 - Q) n< 1 dt biologically unrealistic Subtractive normalization: dw = m (vu – a v2w), a>0 dt Multiplicative normalization:

  21. Multiplicative normalization: dw = m (vu – a v2w), a>0 Multiplicative: dt (Oja’s rule, 1982) This normalization leads to an asymptotic convergence of |w|2 to 1/a. It requires only local information (pre-, post-syn. activity and the local synaptic weight).

  22. Mechanisms of stabilization dw = m vu (v - Q) m << 1 Sliding threshold: biologically unrealistic dt dQ = n (v2 - Q) n< 1 dt biologically unrealistic Subtractive normalization: dw = m (vu – a v2w), a>0 dt no biological evidence Multiplicative normalization: Synaptic Scaling (Turrigiano et al., 1998) could be a candidate to stabilize plasticity

  23. Synaptic Scaling Firing rate homeostasis in cultured networks Block or raise activity (for 2 days) Change of EPSP amplitudes Control After washing out After TTX Response increased Block neuronal activity After Bicuculline Increase neuronal act. Response decreased Long-lasting alterations of neuronal activity induces some changes of the basic properties. Turrigiano and Co-workers, 1998, 2004, 2008,…

  24. Synaptic Scaling Firing rate homeostasis in cultured networks Block or raise activity (for 2 days) Change of EPSP amplitudes Change of EPSP amplitudes Control After washing out After TTX After TTX Response increased Block neuronal activity After Bicuculline After Bicuculline Increase neuronal act. Response decreased Long-lasting alterations of neuronal activity induces some changes of the basic properties. Turrigiano and Co-workers, 1998, 2004, 2008,…

  25. Synaptic Scaling Change of EPSP amplitudes Change of the synaptic weight After TTX Decreased activity (after washing out) After Bicuculline Increased activity Turrigiano and Co-workers, 1998, 2004, 2008,…

  26. Synaptic Scaling Decreased activity Firing rate [v] (after washing out) Increased activity Synaptic weight [w] Turrigiano and Co-workers, 1998, 2004, 2008,…

  27. Synaptic Scaling Synaptic scaling induces stable weight dynamics Theoretical formulation Firing rate [v] Synaptic weight [w] Does synaptic scaling balance the divergent dynamics of synaptic plasticity?

  28. Hebbian Plasticity and Scaling + Hebbian Plasticity Synaptic scaling Hebbian Plasticity ω = μ u v+ γ [vT – v] ω2 ω=μ u v unstable stable

  29. Synaptic Plasticity: STDP LTP Neuron A Neuron B Synapse u v ω LTD Makram et al., 1997 Bi and Poo, 2001

  30. Synaptic Plasticity: Diversity Kampa et al., 2007

  31. Synaptic plasticity together with synaptic scaling is globally stable regardless of learning rule and neuron model. Synaptic scaling is an appropriate candidate to guarantee stability.

  32. Formation of Memory Representations Synaptic Scaling Long-term plasticity Learning Stimulus Input Data from Tetzlaff et al., 2013 Mean Activity Memory Re- presentation of Stimulus Mean Syn. Weight Time

  33. Formation of Memory Representations Synaptic Scaling Long-term plasticity Recall Stimulus Input Data from Tetzlaff et al., 2013 Mean Activity Memory Re- presentation of Stimulus Mean Syn. Weight Time Hebbian Cell Assembly: A group of strongly interconnected neurons, which tend to fire together, form a memory representation of a stimulus. James, 1890; Hebb, 1949; Martin et al., 2000; Martin and Morris, 2002; …

  34. Another learning mechanism Structural Plasticity

  35. Structural plasticity of axon terminals In-vivo 2 Photon-Laser Imaging from the cortex of living mice reveals a permanent axonal remodelling even in the adult brain leading to synaptic rewiring Top: Axonal outgrowth/retraction Red arrow: Axonal outgrowth Yellow: Remains Blue: Retracts Bottom: Rewiring Blue: Retracts and looses synapse Red: Grows and creates new syn.

  36. Spine growth precedes synapse formation Knott et al., 2006 Structural plasticity of dendritic spines Dendritic spines 2μm Dendritic spines on an appical dendrite on a LIV-neuron in V1 of the rat Arellano et al., 2007 Spine formation via filopodia-shaped spines (see arrow, top figure) precedes synapse formation. Spines in synapses are rather mushroom-shaped and carry receptor plates (active zones, red, top figure). Spines contact axonal terminals or axonal varicosities in reach and form synapses (left).

  37. Stable and transient spines In-vivo imaging of dendritic trees within the barrel cortex of living rats Trachtenberg et al., 2002 Structural plasticity of dendritic spines Spines are highly flexible structures that are responsible (together with axonal varicosities) for synaptic rewiring. Only one third of all spines are stable for more than a month. Another third is semistable, meaning that it is present for a couple of days. Transient spines appear and disappear within a day.

  38. Structural Plasticity: More abstract Structural plasticity creates and deletes synapses

  39. Structural Plasticity and Learning training after before camera poor mice structural changes Xu et al., 2009; Yang et al., 2009; Ziv and Ahissar 2009

  40. Structural Plasticity and Learning new spine formation old spine elimination training duration new training task Xu et al., 2009; Yang et al., 2009; Ziv and Ahissar 2009

  41. Structural plasticity monocular deprivation Dendrite of an adult mouse in the visual cortex Hofer et al., 2009 low activity high activity Dendritic Outgrowth Dendritic Regression Kater et al., 1989; Mattson and Kater, 1989

  42. Neuronalactivity Neuronalmorphology Morphology Networkconnectivity . Activity shapes neuronal form and connectivity Connectivity Neuronal activity changes the intracellular calcium. Via changes in intra-cellular calcium, neurons change their morphology with respect to their axonal and dendritic shape. This leads to changes in neuronal connectivity which, in turn, adapts neuronal activity. The goal is that by these changes neurons achieve a homeostatic equilibrium of their activity.

  43. A modelofstructuralplasticity Calcium dependent dendritic andaxonal growth or shrinkage for synapse generation or deletion Circular axonal and dendritic probability spaces overlap ≈ # synapses Overlapdeterminesthesynapticconnectivity. Individual shrinkage and growth following a homeostatic principle. # dendrites: # axons:

  44. The model (withouttheequations) Normal Input Less Input Less Synapses at THIS neuron Homeostatic Process Axonal Growth & Dendr. Shrinkage High Input High Calcium

  45. Structural plasticity model How to define in which developmental stage is a neuronal network? ? Using self-organized criticality Model the development of the neuronal topology

  46. What is self-organized criticality (SOC)? • e.g. the Ising model of magnetism TC Curie temperature ferromagnetic critical paramagnetic • Chialvo, 2007 How to distinguish between states in a dynamic system? “critical” is a state between two different phases of a system.

  47. What is an avalanche? Bak, 1996, How Nature works: The Science of Self-Organized Criticality Now you have a scientific excuse for going to the beach. • An avalanche is an object of spatial-temporal linked events. • Size – number of events within one avalanche • Duration – time between first and last event of the avalanche

  48. What is self-organized criticality (SOC)? • “critical” is a state between two different phases of a system. • Properties of SOC: • The critical state is very robust against external perturbations and responds with avalanches • These avalanches are power law distributed in size and duration • Avalanches are invariant in size and duration: p(ax) = C(a) p(a) “self-organized” describes the property of a system to develop towards a state driven by local rules.

  49. Self-Organized Criticality (a long chapter in Physics) Critical state –> power law –> linear in loglog-plot log(P(size)) log(size) • Levina et al., 2007 log(size) log(size) • An avalanche is an object of spatial-temporally linked events. • Size – number of events within one avalanche • Duration – time between first and last event of the avalanche

  50. Neuronal networks are critical. Beggs and Plenz, 2003 Avalanches and SOC in neuronal networks bin width ∆t

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