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. 2017 Jul 25:6:1222.
doi: 10.12688/f1000research.12130.2. eCollection 2017.

Logarithmic distributions prove that intrinsic learning is Hebbian

Gabriele Scheler  1
Affiliations

Logarithmic distributions prove that intrinsic learning is Hebbian

Gabriele Scheler. F1000Res. .

Abstract

In this paper, we present data for the lognormal distributions of spike rates, synaptic weights and intrinsic excitability (gain) for neurons in various brain areas, such as auditory or visual cortex, hippocampus, cerebellum, striatum, midbrain nuclei. We find a remarkable consistency of heavy-tailed, specifically lognormal, distributions for rates, weights and gains in all brain areas examined. The difference between strongly recurrent and feed-forward connectivity (cortex vs. striatum and cerebellum), neurotransmitter (GABA (striatum) or glutamate (cortex)) or the level of activation (low in cortex, high in Purkinje cells and midbrain nuclei) turns out to be irrelevant for this feature. Logarithmic scale distribution of weights and gains appears to be a general, functional property in all cases analyzed. We then created a generic neural model to investigate adaptive learning rules that create and maintain lognormal distributions. We conclusively demonstrate that not only weights, but also intrinsic gains, need to have strong Hebbian learning in order to produce and maintain the experimentally attested distributions. This provides a solution to the long-standing question about the type of plasticity exhibited by intrinsic excitability.

Keywords: Hebbian learning; intrinsic excitability; lognormal distributions.; neural circuits; neural coding; neural networks; rate coding; spike frequency; synaptic weights.

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Conflict of interest statement

Competing interests: No competing interests were disclosed.

Figures

Figure 1.
Figure 1.. Spike rate data for neurons from inferotemporal cortex (IT) in monkeys .
100 neurons, passive viewing of 77 stimuli, 10 trials (770 data points per neuron), data collected over 200ms. The data are shown for each neuron, where neurons are sorted by mean spike count. A: Mean spike rates (blue), standard deviation (red), and minimum/maximum absolute values (green). B: Mean spike rates histogram shows a lognormal distribution ( σ*=2.32). C: Distributions of standard deviation (green) and CV (blue) have linear slopes, with small variation. The Fano factor (red), measuring the dispersion for each neuron, is nearly constant at about 2.
Figure 2.
Figure 2.. Responses to pure tones in primary auditory cortex from awake monkeys and firing rates from primary auditory cortex in rats .
A: Distribution of spike rates to a 50ms tone ( n = 119, red) 100ms tone ( n = 115, blue), or 200ms tone ( n = 23, green) in primary auditory cortex in monkeys . B: Histogram for spike rate distribution for the 100ms tone response ( n = 115) fitted by an exponential (red) or lognormal (blue) distribution . C: Spontaneous spike rate distribution from primary auditory cortex in unanesthetized rats fitted by an exponential (red) or lognormal (blue) distribution. Note that the spontaneous firing rates are much lower and narrower distributed than evoked spikes in response to stimuli at short time scales ( B), but that they still follow a lognormal distribution.
Figure 3.
Figure 3.. Neuronal response to binaural stimulation for inferior colliculus of the guinea pig ( n = 30), data collected over 100 ms, 200–500 trials.
The data are shown for each neuron, with neurons sorted by mean spike count. A: Mean spike rates (blue), standard deviation (red), and minimum/maximum absolute values (green). B: Mean spike rates histogram shows a lognormal distribution. C: Distributions of standard deviation (green), CV (blue) and FF (red). Again, the dispersion is fairly constant.
Figure 4.
Figure 4.. Spike rates for cerebellar Purkinje cells from rats – .
A: Data for single spikes for Purkinje cells recorded from anesthetized rats (spontaneous in vivo) ( n = 346). B: Data for spike frequencies of isolated cell bodies of mouse Purkinje cells in vitro ( n = 34). C: Spontaneous spike rates for Purkinje cells in slices ( n = 106) . D: Spike counts per neuron from ( n = 34), together with variability data from ( n = 2).
Figure 5.
Figure 5.. Spike rate and gain distributions in basal ganglia .
A: Spike rate in response to a 300ms constant current pulse at 180pA (blue), 200pA (green), and 220pA (red) for neurons in dorsal striatum ( n = 28, solid lines) and nucleus accumbens shell ( n = 24, dashed lines). B: Gain (Hz/nA) for neurons in dorsal striatum ( n = 28). C: Gain (Hz/nA) for neurons in NAcb shell ( n = 24).
Figure 6.
Figure 6.. Gain [Hz/nA] for cortical and striatal neurons.
A: Gain for all types of cortical neurons in vivo ( n = 220) . B: Gain for fast spiking cortical neurons only ( n = 33) . C: Gain for neurons in globus pallidus (GP) in response to a +100pA current pulse ( n = 145) .
Figure 7.
Figure 7.. Strengths of EPSPs in cortex, hippocampus and cerebellum , , , , .
A: Cortex: Deep-layer (L5) pyramidal-pyramidal cell connections , . B: Cortex: Deep-layer (L5) pyramidal-pyramidal cell connections . C: Hippocampus: CA1 to CA3 connections . D: Cerebellum: Granule cells to Purkinje cells .
Figure 8.
Figure 8.. AMPA subunit distribution as a marker of synaptic weight.
A: Expression of labeled GluA1 AMPA receptor subunit in layer 2/3 mouse barrel cortex in vivo follows a lognormal distribution ( σ* = 2.59, µ* = 0.32, n = 560). B: GluA1 density for control (black) and after 1 hour whisker stimulation (red). Stimulation leads to an increase of GluA1 in ≈ 30% of neurons .
Figure 9.
Figure 9.. Generic neural network model with neuron populations I, J (excitatory, blue arrows) and H (inhibitory, red arrows).
Lognormal distributions occur for gain G, rates R I, R J, R H, and weight distributions W IJ. J may have recurrent connectivity.
Figure 10.
Figure 10.. The width of the rate distribution for J, σRJ*, depends heavily on the gain σG*, but not on the weight distribution σW*.
There is a slight effect of connectivity (upper sheet C=5%, lower sheet C=10%). (μW*=0.7,μG*=30,N=1000,σRI*=2.74,μRI*=4.5.)
Figure 11.
Figure 11.. The width of the rate distribution for J, σRJ*, does not depend on R I or R H.
(μW*=0.7,μG*=30,C=10%,N=1000,μRI*=4.5.)
Figure 12.
Figure 12.. Hebbian learning results in lognormal weight distribution independent of gain distribution.
Given is a lognormal input rate σRI*=2,μRI*=4.95. A: Initial weight configurations: Gaussian (grey) or uniform (blue). B: After Hebbian learning using Gaussian gain distribution (grey, blue as before). C: After Hebbian learning using uniform gain distribution (grey, blue as before). D: After Hebbian learning using lognormal gain distribution. (σG*=1.37,μG*=32.7) (grey, blue as before).
Figure 13.
Figure 13.. Homeostatic learning results in Gaussian weight distributions, independent of gain distribution.
Given is a lognormal input rate σRI*=2,μRI*=4.95. A: Initial Configuration: Gaussian (grey) or uniform (blue). B: Homeostatic weight learning using Gaussian gain distribution (grey, blue as before). C: Homeostatic weight learning using uniform gain distribution (grey, blue as before). D: Homeostatic weight learning using lognormal gain distribution (σG*=1.37,μG*=32.7) (grey, blue as before).
Figure 14.
Figure 14.. Hebbian or homeostatic gain learning determine lognormal or Gaussian outcome.
Given is a lognormal input rate σRI*=2,μRI*=4.95. A: Initial Configuration: Gaussian (grey) or uniform (blue). B and C: Hebbian learning using lognormal or Gaussian weights, resulting gain distribution is lognormal. D and E: Homeostatic learning using lognormal or Gaussian weights, resulting gain distribution is Gaussian.
Figure 15.
Figure 15.. Experimental and generated distribution widths for spike rates, gains and weights.
Grey, experimental measurements (s. Tables); red, generated with 100% Hebbian learning; or blue, 80% Hebbian and 20% homeostatic learning combined. The basic distinction in distribution width for gains, rates, weights is reproduced with Hebbian learning, additional homeostatic learning matches experimental values best.
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